Calculus Examples

$\int \sqrt{4 - 5 x^{2}} d x$

Step 1

Let $x = \frac{2}{\sqrt{5}} \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{2 \sqrt{5} \cos (t)}{5} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{2 \sqrt{5} \cos (t)}{5}$ is positive.

$\int \sqrt{4 - 5 {(\frac{2}{\sqrt{5}} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{4 - 5 {(\frac{2}{\sqrt{5}} \sin (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\int \sqrt{4 - 5 {(\frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.2

Combine and simplify the denominator.

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Step 2.1.1.2.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\int \sqrt{4 - 5 {(\frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.2.2

Raise $\sqrt{5}$ to the power of $1$ .

$\int \sqrt{4 - 5 {(\frac{2 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.2.3

Raise $\sqrt{5}$ to the power of $1$ .

$\int \sqrt{4 - 5 {(\frac{2 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.2.4

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

$\int \sqrt{4 - 5 {(\frac{2 \sqrt{5}}{{\sqrt{5}}^{1 + 1}} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.2.5

Add $1$ and $1$ .

$\int \sqrt{4 - 5 {(\frac{2 \sqrt{5}}{{\sqrt{5}}^{2}} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.2.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.1.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\int \sqrt{4 - 5 {(\frac{2 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.2.6.2

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

$\int \sqrt{4 - 5 {(\frac{2 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.2.6.3

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

$\int \sqrt{4 - 5 {(\frac{2 \sqrt{5}}{5^{\frac{2}{2}}} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.1.1.2.6.4.1

Cancel the common factor.

$\int \sqrt{4 - 5 {(\frac{2 \sqrt{5}}{5^{\frac{2}{2}}} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.2.6.4.2

Rewrite the expression.

$\int \sqrt{4 - 5 {(\frac{2 \sqrt{5}}{5^{1}} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.2.6.5

Evaluate the exponent.

$\int \sqrt{4 - 5 {(\frac{2 \sqrt{5}}{5} \sin (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.3

Combine $\frac{2 \sqrt{5}}{5}$ and $\sin (t)$ .

$\int \sqrt{4 - 5 {(\frac{2 \sqrt{5} \sin (t)}{5})}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1.4.1

Apply the product rule to $\frac{2 \sqrt{5} \sin (t)}{5}$ .

$\int \sqrt{4 - 5 \frac{{(2 \sqrt{5} \sin (t))}^{2}}{5^{2}}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.4.2

Apply the product rule to $2 \sqrt{5} \sin (t)$ .

$\int \sqrt{4 - 5 \frac{{(2 \sqrt{5})}^{2} \sin^{2} (t)}{5^{2}}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.4.3

Apply the product rule to $2 \sqrt{5}$ .

$\int \sqrt{4 - 5 \frac{2^{2} {\sqrt{5}}^{2} \sin^{2} (t)}{5^{2}}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.5

Simplify the numerator.

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Step 2.1.1.5.1

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

$\int \sqrt{4 - 5 \frac{4 {\sqrt{5}}^{2} \sin^{2} (t)}{5^{2}}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.5.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.1.1.5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\int \sqrt{4 - 5 \frac{4 {(5^{\frac{1}{2}})}^{2} \sin^{2} (t)}{5^{2}}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.5.2.2

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

$\int \sqrt{4 - 5 \frac{4 \cdot 5^{\frac{1}{2} \cdot 2} \sin^{2} (t)}{5^{2}}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.5.2.3

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

$\int \sqrt{4 - 5 \frac{4 \cdot 5^{\frac{2}{2}} \sin^{2} (t)}{5^{2}}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.5.2.4

Cancel the common factor of $2$ .

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Step 2.1.1.5.2.4.1

Cancel the common factor.

$\int \sqrt{4 - 5 \frac{4 \cdot 5^{\frac{2}{2}} \sin^{2} (t)}{5^{2}}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.5.2.4.2

Rewrite the expression.

$\int \sqrt{4 - 5 \frac{4 \cdot 5^{1} \sin^{2} (t)}{5^{2}}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.5.2.5

Evaluate the exponent.

$\int \sqrt{4 - 5 \frac{4 \cdot 5 \sin^{2} (t)}{5^{2}}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.5.3

Multiply $4$ by $5$ .

$\int \sqrt{4 - 5 \frac{20 \sin^{2} (t)}{5^{2}}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.6

Raise $5$ to the power of $2$ .

$\int \sqrt{4 - 5 \frac{20 \sin^{2} (t)}{25}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.7

Cancel the common factor of $5$ .

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Step 2.1.1.7.1

Factor $5$ out of $- 5$ .

$\int \sqrt{4 + 5 (- 1) \frac{20 \sin^{2} (t)}{25}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.7.2

Factor $5$ out of $25$ .

$\int \sqrt{4 + 5 \cdot - 1 \frac{20 \sin^{2} (t)}{5 \cdot 5}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.7.3

Cancel the common factor.

$\int \sqrt{4 + 5 \cdot - 1 \frac{20 \sin^{2} (t)}{5 \cdot 5}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.7.4

Rewrite the expression.

$\int \sqrt{4 - 1 \frac{20 \sin^{2} (t)}{5}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.8

Cancel the common factor of $20$ and $5$ .

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Step 2.1.1.8.1

Factor $5$ out of $20 \sin^{2} (t)$ .

$\int \sqrt{4 - 1 \frac{5 (4 \sin^{2} (t))}{5}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.8.2

Cancel the common factors.

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Step 2.1.1.8.2.1

Factor $5$ out of $5$ .

$\int \sqrt{4 - 1 \frac{5 (4 \sin^{2} (t))}{5 (1)}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.8.2.2

Cancel the common factor.

$\int \sqrt{4 - 1 \frac{5 (4 \sin^{2} (t))}{5 \cdot 1}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.8.2.3

Rewrite the expression.

$\int \sqrt{4 - 1 \frac{4 \sin^{2} (t)}{1}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.8.2.4

Divide $4 \sin^{2} (t)$ by $1$ .

$\int \sqrt{4 - 1 (4 \sin^{2} (t))} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.1.9

Multiply $4$ by $- 1$ .

$\int \sqrt{4 - 4 \sin^{2} (t)} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.2

Factor $4$ out of $4$ .

$\int \sqrt{4 (1) - 4 \sin^{2} (t)} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.3

Factor $4$ out of $- 4 \sin^{2} (t)$ .

$\int \sqrt{4 (1) + 4 (- \sin^{2} (t))} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.4

Factor $4$ out of $4 (1) + 4 (- \sin^{2} (t))$ .

$\int \sqrt{4 (1 - \sin^{2} (t))} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.5

Apply pythagorean identity.

$\int \sqrt{4 \cos^{2} (t)} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.6

Rewrite $4 \cos^{2} (t)$ as ${(2 \cos (t))}^{2}$ .

$\int \sqrt{{(2 \cos (t))}^{2}} \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.1.7

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

$\int 2 \cos (t) \frac{2 \sqrt{5} \cos (t)}{5} d t$

Step 2.2

Simplify.

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Step 2.2.1

Combine $\frac{2 \sqrt{5} \cos (t)}{5}$ and $2$ .

$\int \frac{2 \sqrt{5} \cos (t) \cdot 2}{5} \cos (t) d t$

Step 2.2.2

Multiply $2$ by $2$ .

$\int \frac{4 \sqrt{5} \cos (t)}{5} \cos (t) d t$

Step 2.2.3

Combine $\frac{4 \sqrt{5} \cos (t)}{5}$ and $\cos (t)$ .

$\int \frac{4 \sqrt{5} \cos (t) \cos (t)}{5} d t$

Step 2.2.4

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

$\int \frac{4 \sqrt{5} (\cos^{1} (t) \cos (t))}{5} d t$

Step 2.2.5

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

$\int \frac{4 \sqrt{5} (\cos^{1} (t) \cos^{1} (t))}{5} d t$

Step 2.2.6

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

$\int \frac{4 \sqrt{5} {\cos (t)}^{1 + 1}}{5} d t$

Step 2.2.7

Add $1$ and $1$ .

$\int \frac{4 \sqrt{5} \cos^{2} (t)}{5} d t$

Step 3

Since $\frac{4 \sqrt{5}}{5}$ is constant with respect to $t$ , move $\frac{4 \sqrt{5}}{5}$ out of the integral.

$\frac{4 \sqrt{5}}{5} \int \cos^{2} (t) d t$

Step 4

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

$\frac{4 \sqrt{5}}{5} \int \frac{1 + \cos (2 t)}{2} d t$

Step 5

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

$\frac{4 \sqrt{5}}{5} (\frac{1}{2} \int 1 + \cos (2 t) d t)$

Step 6

Simplify.

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Step 6.1

Multiply $\frac{1}{2}$ by $\frac{4 \sqrt{5}}{5}$ .

$\frac{4 \sqrt{5}}{2 \cdot 5} \int 1 + \cos (2 t) d t$

Step 6.2

Multiply $2$ by $5$ .

$\frac{4 \sqrt{5}}{10} \int 1 + \cos (2 t) d t$

Step 6.3

Cancel the common factor of $4$ and $10$ .

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Step 6.3.1

Factor $2$ out of $4 \sqrt{5}$ .

$\frac{2 (2 \sqrt{5})}{10} \int 1 + \cos (2 t) d t$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $2$ out of $10$ .

$\frac{2 (2 \sqrt{5})}{2 (5)} \int 1 + \cos (2 t) d t$

Step 6.3.2.2

Cancel the common factor.

$\frac{2 (2 \sqrt{5})}{2 \cdot 5} \int 1 + \cos (2 t) d t$

Step 6.3.2.3

Rewrite the expression.

$\frac{2 \sqrt{5}}{5} \int 1 + \cos (2 t) d t$

Step 7

Split the single integral into multiple integrals.

$\frac{2 \sqrt{5}}{5} (\int d t + \int \cos (2 t) d t)$

Step 8

Apply the constant rule.

$\frac{2 \sqrt{5}}{5} (t + C + \int \cos (2 t) d t)$

Step 9

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

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Step 9.1

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

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Step 9.1.1

Differentiate $2 t$ .

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

Step 9.1.2

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 9.1.3

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 9.1.4

Multiply $2$ by $1$ .

$2$

Step 9.2

Rewrite the problem using $u$ and $d u$ .

$\frac{2 \sqrt{5}}{5} (t + C + \int \cos (u) \frac{1}{2} d u)$

Step 10

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

$\frac{2 \sqrt{5}}{5} (t + C + \int \frac{\cos (u)}{2} d u)$

Step 11

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

$\frac{2 \sqrt{5}}{5} (t + C + \frac{1}{2} \int \cos (u) d u)$

Step 12

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{2 \sqrt{5}}{5} (t + C + \frac{1}{2} (\sin (u) + C))$

Step 13

Simplify.

$\frac{2 \sqrt{5}}{5} (t + \frac{1}{2} \sin (u)) + C$

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $t$ with $\arcsin (\frac{\sqrt{5} x}{2})$ .

$\frac{2 \sqrt{5}}{5} (\arcsin (\frac{\sqrt{5} x}{2}) + \frac{1}{2} \sin (u)) + C$

Step 14.2

Replace all occurrences of $u$ with $2 t$ .

$\frac{2 \sqrt{5}}{5} (\arcsin (\frac{\sqrt{5} x}{2}) + \frac{1}{2} \sin (2 t)) + C$

Step 14.3

Replace all occurrences of $t$ with $\arcsin (\frac{\sqrt{5} x}{2})$ .

$\frac{2 \sqrt{5}}{5} (\arcsin (\frac{\sqrt{5} x}{2}) + \frac{1}{2} \sin (2 \arcsin (\frac{\sqrt{5} x}{2}))) + C$

Step 15

Simplify.

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Step 15.1

Combine $\frac{1}{2}$ and $\sin (2 \arcsin (\frac{\sqrt{5} x}{2}))$ .

$\frac{2 \sqrt{5}}{5} (\arcsin (\frac{\sqrt{5} x}{2}) + \frac{\sin (2 \arcsin (\frac{\sqrt{5} x}{2}))}{2}) + C$

Step 15.2

Apply the distributive property.

$\frac{2 \sqrt{5}}{5} \arcsin (\frac{\sqrt{5} x}{2}) + \frac{2 \sqrt{5}}{5} \cdot \frac{\sin (2 \arcsin (\frac{\sqrt{5} x}{2}))}{2} + C$

Step 15.3

Combine $\frac{2 \sqrt{5}}{5}$ and $\arcsin (\frac{\sqrt{5} x}{2})$ .

$\frac{2 \sqrt{5} \arcsin (\frac{\sqrt{5} x}{2})}{5} + \frac{2 \sqrt{5}}{5} \cdot \frac{\sin (2 \arcsin (\frac{\sqrt{5} x}{2}))}{2} + C$

Step 15.4

Cancel the common factor of $2$ .

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Step 15.4.1

Factor $2$ out of $2 \sqrt{5}$ .

$\frac{2 \sqrt{5} \arcsin (\frac{\sqrt{5} x}{2})}{5} + \frac{2 (\sqrt{5})}{5} \cdot \frac{\sin (2 \arcsin (\frac{\sqrt{5} x}{2}))}{2} + C$

Step 15.4.2

Cancel the common factor.

$\frac{2 \sqrt{5} \arcsin (\frac{\sqrt{5} x}{2})}{5} + \frac{2 \sqrt{5}}{5} \cdot \frac{\sin (2 \arcsin (\frac{\sqrt{5} x}{2}))}{2} + C$

Step 15.4.3

Rewrite the expression.

$\frac{2 \sqrt{5} \arcsin (\frac{\sqrt{5} x}{2})}{5} + \frac{\sqrt{5}}{5} \sin (2 \arcsin (\frac{\sqrt{5} x}{2})) + C$

Step 15.5

Combine $\frac{\sqrt{5}}{5}$ and $\sin (2 \arcsin (\frac{\sqrt{5} x}{2}))$ .

$\frac{2 \sqrt{5} \arcsin (\frac{\sqrt{5} x}{2})}{5} + \frac{\sqrt{5} \sin (2 \arcsin (\frac{\sqrt{5} x}{2}))}{5} + C$

Step 16

Reorder terms.

$\frac{2 \sqrt{5}}{5} \arcsin (\frac{\sqrt{5}}{2} x) + \frac{\sqrt{5}}{5} \sin (2 \arcsin (\frac{\sqrt{5}}{2} x)) + C$