Calculus Examples

$\int {(16 - x^{2})}^{\frac{3}{2}} d x$

Step 1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\int \sqrt{{(16 - x^{2})}^{3}} d x$

Step 2

Let $x = 4 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 4 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $4 \cos (t)$ is positive.

$\int \sqrt{{(16 - {(4 \sin (t))}^{2})}^{3}} (4 \cos (t)) d t$

Step 3

Simplify terms.

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Step 3.1

Simplify $\sqrt{{(16 - {(4 \sin (t))}^{2})}^{3}}$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the product rule to $4 \sin (t)$ .

$\int \sqrt{{(16 - (4^{2} \sin^{2} (t)))}^{3}} (4 \cos (t)) d t$

Step 3.1.1.2

Raise $4$ to the power of $2$ .

$\int \sqrt{{(16 - (16 \sin^{2} (t)))}^{3}} (4 \cos (t)) d t$

Step 3.1.1.3

Multiply $16$ by $- 1$ .

$\int \sqrt{{(16 - 16 \sin^{2} (t))}^{3}} (4 \cos (t)) d t$

Step 3.1.2

Factor $16$ out of $16$ .

$\int \sqrt{{(16 (1) - 16 \sin^{2} (t))}^{3}} (4 \cos (t)) d t$

Step 3.1.3

Factor $16$ out of $- 16 \sin^{2} (t)$ .

$\int \sqrt{{(16 (1) + 16 (- \sin^{2} (t)))}^{3}} (4 \cos (t)) d t$

Step 3.1.4

Factor $16$ out of $16 (1) + 16 (- \sin^{2} (t))$ .

$\int \sqrt{{(16 (1 - \sin^{2} (t)))}^{3}} (4 \cos (t)) d t$

Step 3.1.5

Apply pythagorean identity.

$\int \sqrt{{(16 \cos^{2} (t))}^{3}} (4 \cos (t)) d t$

Step 3.1.6

Apply the product rule to $16 \cos^{2} (t)$ .

$\int \sqrt{16^{3} {(\cos^{2} (t))}^{3}} (4 \cos (t)) d t$

Step 3.1.7

Raise $16$ to the power of $3$ .

$\int \sqrt{4096 {(\cos^{2} (t))}^{3}} (4 \cos (t)) d t$

Step 3.1.8

Multiply the exponents in ${(\cos^{2} (t))}^{3}$ .

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Step 3.1.8.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int \sqrt{4096 {\cos (t)}^{2 \cdot 3}} (4 \cos (t)) d t$

Step 3.1.8.2

Multiply $2$ by $3$ .

$\int \sqrt{4096 \cos^{6} (t)} (4 \cos (t)) d t$

Step 3.1.9

Rewrite $4096 \cos^{6} (t)$ as ${(64 \cos^{3} (t))}^{2}$ .

$\int \sqrt{{(64 \cos^{3} (t))}^{2}} (4 \cos (t)) d t$

Step 3.1.10

Pull terms out from under the radical, assuming positive real numbers.

$\int 64 \cos^{3} (t) (4 \cos (t)) d t$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $4$ by $64$ .

$\int 256 \cos^{3} (t) \cos (t) d t$

Step 3.2.2

Raise $\cos (t)$ to the power of $1$ .

$\int 256 (\cos^{1} (t) \cos^{3} (t)) d t$

Step 3.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 256 {\cos (t)}^{1 + 3} d t$

Step 3.2.4

Add $1$ and $3$ .

$\int 256 \cos^{4} (t) d t$

Step 4

Since $256$ is constant with respect to $t$ , move $256$ out of the integral.

$256 \int \cos^{4} (t) d t$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $2$ out of $4$ .

$256 \int {\cos (t)}^{2 (2)} d t$

Step 5.2

Rewrite ${\cos (t)}^{2 (2)}$ as exponentiation.

$256 \int {(\cos^{2} (t))}^{2} d t$

Step 6

Use the half-angle formula to rewrite $\cos^{2} (t)$ as $\frac{1 + \cos (2 t)}{2}$ .

$256 \int {(\frac{1 + \cos (2 t)}{2})}^{2} d t$

Step 7

Let $u_{1} = 2 t$ . Then $d u_{1} = 2 d t$ , so $\frac{1}{2} d u_{1} = d t$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 7.1

Let $u_{1} = 2 t$ . Find $\frac{d u_{1}}{d t}$ .

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Step 7.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 7.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

Step 7.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 7.1.4

Multiply $2$ by $1$ .

$2$

Step 7.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$256 \int {(\frac{1 + \cos (u_{1})}{2})}^{2} \frac{1}{2} d u_{1}$

Step 8

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$256 (\frac{1}{2} \int {(\frac{1 + \cos (u_{1})}{2})}^{2} d u_{1})$

Step 9

Simplify terms.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $\frac{1}{2}$ and $256$ .

$\frac{256}{2} \int {(\frac{1 + \cos (u_{1})}{2})}^{2} d u_{1}$

Step 9.1.2

Cancel the common factor of $256$ and $2$ .

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Step 9.1.2.1

Factor $2$ out of $256$ .

$\frac{2 \cdot 128}{2} \int {(\frac{1 + \cos (u_{1})}{2})}^{2} d u_{1}$

Step 9.1.2.2

Cancel the common factors.

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Step 9.1.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 128}{2 (1)} \int {(\frac{1 + \cos (u_{1})}{2})}^{2} d u_{1}$

Step 9.1.2.2.2

Cancel the common factor.

$\frac{2 \cdot 128}{2 \cdot 1} \int {(\frac{1 + \cos (u_{1})}{2})}^{2} d u_{1}$

Step 9.1.2.2.3

Rewrite the expression.

$\frac{128}{1} \int {(\frac{1 + \cos (u_{1})}{2})}^{2} d u_{1}$

Step 9.1.2.2.4

Divide $128$ by $1$ .

$128 \int {(\frac{1 + \cos (u_{1})}{2})}^{2} d u_{1}$

Step 9.2

Rewrite $\frac{1 + \cos (u_{1})}{2}$ as a product.

$128 \int {(\frac{1}{2} \cdot (1 + \cos (u_{1})))}^{2} d u_{1}$

Step 9.3

Expand ${(\frac{1}{2} \cdot (1 + \cos (u_{1})))}^{2}$ .

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Step 9.3.1

Rewrite the exponentiation as a product.

$128 \int \frac{1}{2} \cdot (1 + \cos (u_{1})) (\frac{1}{2} \cdot (1 + \cos (u_{1}))) d u_{1}$

Step 9.3.2

Apply the distributive property.

$128 \int (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{1})) (\frac{1}{2} \cdot (1 + \cos (u_{1}))) d u_{1}$

Step 9.3.3

Apply the distributive property.

$128 \int (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{1})) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{1})) d u_{1}$

Step 9.3.4

Apply the distributive property.

$128 \int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{1})) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{1})) d u_{1}$

Step 9.3.5

Apply the distributive property.

$128 \int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{1})) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot 1 + \frac{1}{2} \cdot \cos (u_{1})) d u_{1}$

Step 9.3.6

Apply the distributive property.

$128 \int \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{1})) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot \cos (u_{1})) d u_{1}$

Step 9.3.7

Reorder $\frac{1}{2}$ and $1$ .

$128 \int 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{1})) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot \cos (u_{1})) d u_{1}$

Step 9.3.8

Reorder $\frac{1}{2}$ and $1$ .

$128 \int 1 \cdot \frac{1}{2} (1 \cdot \frac{1}{2}) + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{1})) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot \cos (u_{1})) d u_{1}$

Step 9.3.9

Move $\frac{1}{2}$ .

$128 \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot 1 (\frac{1}{2} \cdot \cos (u_{1})) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot \cos (u_{1})) d u_{1}$

Step 9.3.10

Reorder $\frac{1}{2}$ and $1$ .

$128 \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} (\frac{1}{2} \cdot \cos (u_{1})) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot 1) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot \cos (u_{1})) d u_{1}$

Step 9.3.11

Reorder $\frac{1}{2}$ and $1$ .

$128 \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{1}) + \frac{1}{2} \cdot \cos (u_{1}) (1 \cdot \frac{1}{2}) + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot \cos (u_{1})) d u_{1}$

Step 9.3.12

Move $\cos (u_{1})$ .

$128 \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{1}) + \frac{1}{2} \cdot 1 \cos (u_{1}) \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot \cos (u_{1})) d u_{1}$

Step 9.3.13

Reorder $\frac{1}{2}$ and $1$ .

$128 \int 1 \cdot 1 \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} \frac{1}{2} \cdot \cos (u_{1}) + 1 \cdot \frac{1}{2} \cos (u_{1}) \cdot \frac{1}{2} + \frac{1}{2} \cdot \cos (u_{1}) (\frac{1}{2} \cdot \cos (u_{1})) d u_{1}$

Step 9.3.14

Multiply $1$ by $1$ .

$128 \int 1 (\frac{1}{2}) \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.15

Multiply $\frac{1}{2}$ by $1$ .

$128 \int \frac{1}{2} \cdot \frac{1}{2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.16

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$128 \int \frac{1}{2 \cdot 2} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.17

Multiply $2$ by $2$ .

$128 \int \frac{1}{4} + 1 (\frac{1}{2}) \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.18

Multiply $\frac{1}{2}$ by $1$ .

$128 \int \frac{1}{4} + \frac{1}{2} \cdot \frac{1}{2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.19

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$128 \int \frac{1}{4} + \frac{1}{2 \cdot 2} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.20

Multiply $2$ by $2$ .

$128 \int \frac{1}{4} + \frac{1}{4} \cos (u_{1}) + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.21

Combine $\frac{1}{4}$ and $\cos (u_{1})$ .

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + 1 (\frac{1}{2}) \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.22

Multiply $\frac{1}{2}$ by $1$ .

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.23

Combine $\frac{1}{2}$ and $\cos (u_{1})$ .

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{2} \cdot \frac{1}{2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.24

Multiply $\frac{\cos (u_{1})}{2}$ by $\frac{1}{2}$ .

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{2 \cdot 2} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.25

Multiply $2$ by $2$ .

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{4} + \frac{1}{2} \cos (u_{1}) \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.26

Combine $\frac{1}{2}$ and $\cos (u_{1})$ .

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{2} \cdot \frac{1}{2} \cos (u_{1}) d u_{1}$

Step 9.3.27

Multiply $\frac{\cos (u_{1})}{2}$ by $\frac{1}{2}$ .

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{2 \cdot 2} \cos (u_{1}) d u_{1}$

Step 9.3.28

Multiply $2$ by $2$ .

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{4} \cos (u_{1}) d u_{1}$

Step 9.3.29

Combine $\frac{\cos (u_{1})}{4}$ and $\cos (u_{1})$ .

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1}) \cos (u_{1})}{4} d u_{1}$

Step 9.3.30

Raise $\cos (u_{1})$ to the power of $1$ .

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos^{1} (u_{1}) \cos (u_{1})}{4} d u_{1}$

Step 9.3.31

Raise $\cos (u_{1})$ to the power of $1$ .

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos^{1} (u_{1}) \cos^{1} (u_{1})}{4} d u_{1}$

Step 9.3.32

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{4} + \frac{{\cos (u_{1})}^{1 + 1}}{4} d u_{1}$

Step 9.3.33

Add $1$ and $1$ .

$128 \int \frac{1}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos (u_{1})}{4} + \frac{\cos^{2} (u_{1})}{4} d u_{1}$

Step 9.3.34

Add $\frac{\cos (u_{1})}{4}$ and $\frac{\cos (u_{1})}{4}$ .

$128 \int \frac{1}{4} + 2 \frac{\cos (u_{1})}{4} + \frac{\cos^{2} (u_{1})}{4} d u_{1}$

Step 9.3.35

Combine $2$ and $\frac{\cos (u_{1})}{4}$ .

$128 \int \frac{1}{4} + \frac{2 \cos (u_{1})}{4} + \frac{\cos^{2} (u_{1})}{4} d u_{1}$

Step 9.3.36

Reorder $\frac{2 \cos (u_{1})}{4}$ and $\frac{\cos^{2} (u_{1})}{4}$ .

$128 \int \frac{1}{4} + \frac{\cos^{2} (u_{1})}{4} + \frac{2 \cos (u_{1})}{4} d u_{1}$

Step 9.3.37

Reorder $\frac{1}{4}$ and $\frac{\cos^{2} (u_{1})}{4}$ .

$128 \int \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{2 \cos (u_{1})}{4} d u_{1}$

Step 9.4

Cancel the common factor of $2$ and $4$ .

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Step 9.4.1

Factor $2$ out of $2 \cos (u_{1})$ .

$128 \int \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{2 (\cos (u_{1}))}{4} d u_{1}$

Step 9.4.2

Cancel the common factors.

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Step 9.4.2.1

Factor $2$ out of $4$ .

$128 \int \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{2 \cos (u_{1})}{2 \cdot 2} d u_{1}$

Step 9.4.2.2

Cancel the common factor.

$128 \int \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{2 \cos (u_{1})}{2 \cdot 2} d u_{1}$

Step 9.4.2.3

Rewrite the expression.

$128 \int \frac{\cos^{2} (u_{1})}{4} + \frac{1}{4} + \frac{\cos (u_{1})}{2} d u_{1}$

Step 10

Split the single integral into multiple integrals.

$128 (\int \frac{\cos^{2} (u_{1})}{4} d u_{1} + \int \frac{1}{4} d u_{1} + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 11

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

$128 (\frac{1}{4} \int \cos^{2} (u_{1}) d u_{1} + \int \frac{1}{4} d u_{1} + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 12

Use the half-angle formula to rewrite $\cos^{2} (u_{1})$ as $\frac{1 + \cos (2 u_{1})}{2}$ .

$128 (\frac{1}{4} \int \frac{1 + \cos (2 u_{1})}{2} d u_{1} + \int \frac{1}{4} d u_{1} + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 13

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$128 (\frac{1}{4} (\frac{1}{2} \int 1 + \cos (2 u_{1}) d u_{1}) + \int \frac{1}{4} d u_{1} + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 14

Simplify.

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Step 14.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$128 (\frac{1}{2 \cdot 4} \int 1 + \cos (2 u_{1}) d u_{1} + \int \frac{1}{4} d u_{1} + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 14.2

Multiply $2$ by $4$ .

$128 (\frac{1}{8} \int 1 + \cos (2 u_{1}) d u_{1} + \int \frac{1}{4} d u_{1} + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 15

Split the single integral into multiple integrals.

$128 (\frac{1}{8} (\int d u_{1} + \int \cos (2 u_{1}) d u_{1}) + \int \frac{1}{4} d u_{1} + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 16

Apply the constant rule.

$128 (\frac{1}{8} (u_{1} + C + \int \cos (2 u_{1}) d u_{1}) + \int \frac{1}{4} d u_{1} + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 17

Let $u_{2} = 2 u_{1}$ . Then $d u_{2} = 2 d u_{1}$ , so $\frac{1}{2} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 17.1

Let $u_{2} = 2 u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 17.1.1

Differentiate $2 u_{1}$ .

$\frac{d}{d u_{1}} [2 u_{1}]$

Step 17.1.2

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$2 \frac{d}{d u_{1}} [u_{1}]$

Step 17.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 17.1.4

Multiply $2$ by $1$ .

$2$

Step 17.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$128 (\frac{1}{8} (u_{1} + C + \int \cos (u_{2}) \frac{1}{2} d u_{2}) + \int \frac{1}{4} d u_{1} + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 18

Combine $\cos (u_{2})$ and $\frac{1}{2}$ .

$128 (\frac{1}{8} (u_{1} + C + \int \frac{\cos (u_{2})}{2} d u_{2}) + \int \frac{1}{4} d u_{1} + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 19

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$128 (\frac{1}{8} (u_{1} + C + \frac{1}{2} \int \cos (u_{2}) d u_{2}) + \int \frac{1}{4} d u_{1} + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 20

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$128 (\frac{1}{8} (u_{1} + C + \frac{1}{2} (\sin (u_{2}) + C)) + \int \frac{1}{4} d u_{1} + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 21

Apply the constant rule.

$128 (\frac{1}{8} (u_{1} + C + \frac{1}{2} (\sin (u_{2}) + C)) + \frac{1}{4} u_{1} + C + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 22

Combine $\frac{1}{4}$ and $u_{1}$ .

$128 (\frac{1}{8} (u_{1} + C + \frac{1}{2} (\sin (u_{2}) + C)) + \frac{u_{1}}{4} + C + \int \frac{\cos (u_{1})}{2} d u_{1})$

Step 23

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$128 (\frac{1}{8} (u_{1} + C + \frac{1}{2} (\sin (u_{2}) + C)) + \frac{u_{1}}{4} + C + \frac{1}{2} \int \cos (u_{1}) d u_{1})$

Step 24

The integral of $\cos (u_{1})$ with respect to $u_{1}$ is $\sin (u_{1})$ .

$128 (\frac{1}{8} (u_{1} + C + \frac{1}{2} (\sin (u_{2}) + C)) + \frac{u_{1}}{4} + C + \frac{1}{2} (\sin (u_{1}) + C))$

Step 25

Simplify.

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Step 25.1

Simplify.

$128 (\frac{u_{1}}{8} + \frac{\sin (u_{2})}{16} + \frac{u_{1}}{4} + \frac{\sin (u_{1})}{2}) + C$

Step 25.2

Simplify.

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Step 25.2.1

To write $\frac{u_{1}}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$128 (\frac{u_{1}}{8} + \frac{u_{1}}{4} \cdot \frac{2}{2} + \frac{\sin (u_{2})}{16} + \frac{\sin (u_{1})}{2}) + C$

Step 25.2.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 25.2.2.1

Multiply $\frac{u_{1}}{4}$ by $\frac{2}{2}$ .

$128 (\frac{u_{1}}{8} + \frac{u_{1} \cdot 2}{4 \cdot 2} + \frac{\sin (u_{2})}{16} + \frac{\sin (u_{1})}{2}) + C$

Step 25.2.2.2

Multiply $4$ by $2$ .

$128 (\frac{u_{1}}{8} + \frac{u_{1} \cdot 2}{8} + \frac{\sin (u_{2})}{16} + \frac{\sin (u_{1})}{2}) + C$

Step 25.2.3

Combine the numerators over the common denominator.

$128 (\frac{u_{1} + u_{1} \cdot 2}{8} + \frac{\sin (u_{2})}{16} + \frac{\sin (u_{1})}{2}) + C$

Step 25.2.4

Move $2$ to the left of $u_{1}$ .

$128 (\frac{u_{1} + 2 \cdot u_{1}}{8} + \frac{\sin (u_{2})}{16} + \frac{\sin (u_{1})}{2}) + C$

Step 25.2.5

Add $u_{1}$ and $2 u_{1}$ .

$128 (\frac{3 u_{1}}{8} + \frac{\sin (u_{2})}{16} + \frac{\sin (u_{1})}{2}) + C$

Step 26

Substitute back in for each integration substitution variable.

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Step 26.1

Replace all occurrences of $u_{1}$ with $2 t$ .

$128 (\frac{3 (2 t)}{8} + \frac{\sin (u_{2})}{16} + \frac{\sin (2 t)}{2}) + C$

Step 26.2

Replace all occurrences of $u_{2}$ with $2 u_{1}$ .

$128 (\frac{3 (2 t)}{8} + \frac{\sin (2 u_{1})}{16} + \frac{\sin (2 t)}{2}) + C$

Step 26.3

Replace all occurrences of $t$ with $\arcsin (\frac{x}{4})$ .

$128 (\frac{3 (2 \arcsin (\frac{x}{4}))}{8} + \frac{\sin (2 u_{1})}{16} + \frac{\sin (2 \arcsin (\frac{x}{4}))}{2}) + C$

Step 26.4

Replace all occurrences of $u_{1}$ with $2 t$ .

$128 (\frac{3 (2 \arcsin (\frac{x}{4}))}{8} + \frac{\sin (2 (2 t))}{16} + \frac{\sin (2 \arcsin (\frac{x}{4}))}{2}) + C$

Step 26.5

Replace all occurrences of $t$ with $\arcsin (\frac{x}{4})$ .

$128 (\frac{3 (2 \arcsin (\frac{x}{4}))}{8} + \frac{\sin (2 (2 \arcsin (\frac{x}{4})))}{16} + \frac{\sin (2 \arcsin (\frac{x}{4}))}{2}) + C$

Step 27

Simplify.

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Step 27.1

Simplify each term.

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Step 27.1.1

Cancel the common factor of $2$ and $8$ .

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Step 27.1.1.1

Factor $2$ out of $3 (2 \arcsin (\frac{x}{4}))$ .

$128 (\frac{2 (3 (\arcsin (\frac{x}{4})))}{8} + \frac{\sin (2 (2 \arcsin (\frac{x}{4})))}{16} + \frac{\sin (2 \arcsin (\frac{x}{4}))}{2}) + C$

Step 27.1.1.2

Cancel the common factors.

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Step 27.1.1.2.1

Factor $2$ out of $8$ .

$128 (\frac{2 (3 (\arcsin (\frac{x}{4})))}{2 \cdot 4} + \frac{\sin (2 (2 \arcsin (\frac{x}{4})))}{16} + \frac{\sin (2 \arcsin (\frac{x}{4}))}{2}) + C$

Step 27.1.1.2.2

Cancel the common factor.

$128 (\frac{2 (3 (\arcsin (\frac{x}{4})))}{2 \cdot 4} + \frac{\sin (2 (2 \arcsin (\frac{x}{4})))}{16} + \frac{\sin (2 \arcsin (\frac{x}{4}))}{2}) + C$

Step 27.1.1.2.3

Rewrite the expression.

$128 (\frac{3 (\arcsin (\frac{x}{4}))}{4} + \frac{\sin (2 (2 \arcsin (\frac{x}{4})))}{16} + \frac{\sin (2 \arcsin (\frac{x}{4}))}{2}) + C$

Step 27.1.2

Multiply $2$ by $2$ .

$128 (\frac{3 \arcsin (\frac{x}{4})}{4} + \frac{\sin (4 \arcsin (\frac{x}{4}))}{16} + \frac{\sin (2 \arcsin (\frac{x}{4}))}{2}) + C$

Step 27.2

Apply the distributive property.

$128 \frac{3 \arcsin (\frac{x}{4})}{4} + 128 \frac{\sin (4 \arcsin (\frac{x}{4}))}{16} + 128 \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 27.3

Simplify.

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Step 27.3.1

Cancel the common factor of $4$ .

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Step 27.3.1.1

Factor $4$ out of $128$ .

$4 (32) \frac{3 \arcsin (\frac{x}{4})}{4} + 128 \frac{\sin (4 \arcsin (\frac{x}{4}))}{16} + 128 \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 27.3.1.2

Cancel the common factor.

$4 \cdot 32 \frac{3 \arcsin (\frac{x}{4})}{4} + 128 \frac{\sin (4 \arcsin (\frac{x}{4}))}{16} + 128 \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 27.3.1.3

Rewrite the expression.

$32 (3 \arcsin (\frac{x}{4})) + 128 \frac{\sin (4 \arcsin (\frac{x}{4}))}{16} + 128 \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 27.3.2

Multiply $3$ by $32$ .

$96 \arcsin (\frac{x}{4}) + 128 \frac{\sin (4 \arcsin (\frac{x}{4}))}{16} + 128 \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 27.3.3

Cancel the common factor of $16$ .

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Step 27.3.3.1

Factor $16$ out of $128$ .

$96 \arcsin (\frac{x}{4}) + 16 (8) \frac{\sin (4 \arcsin (\frac{x}{4}))}{16} + 128 \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 27.3.3.2

Cancel the common factor.

$96 \arcsin (\frac{x}{4}) + 16 \cdot 8 \frac{\sin (4 \arcsin (\frac{x}{4}))}{16} + 128 \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 27.3.3.3

Rewrite the expression.

$96 \arcsin (\frac{x}{4}) + 8 \sin (4 \arcsin (\frac{x}{4})) + 128 \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 27.3.4

Cancel the common factor of $2$ .

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Step 27.3.4.1

Factor $2$ out of $128$ .

$96 \arcsin (\frac{x}{4}) + 8 \sin (4 \arcsin (\frac{x}{4})) + 2 (64) \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 27.3.4.2

Cancel the common factor.

$96 \arcsin (\frac{x}{4}) + 8 \sin (4 \arcsin (\frac{x}{4})) + 2 \cdot 64 \frac{\sin (2 \arcsin (\frac{x}{4}))}{2} + C$

Step 27.3.4.3

Rewrite the expression.

$96 \arcsin (\frac{x}{4}) + 8 \sin (4 \arcsin (\frac{x}{4})) + 64 \sin (2 \arcsin (\frac{x}{4})) + C$

Step 28

Reorder terms.

$96 \arcsin (\frac{1}{4} x) + 8 \sin (4 \arcsin (\frac{1}{4} x)) + 64 \sin (2 \arcsin (\frac{1}{4} x)) + C$