Calculus Examples

$\int_{0}^{\frac{3 \sqrt{3}}{2}} \frac{x^{3}}{\sqrt{4 x^{2} + 9}} d x$

Step 1

Let $x = \frac{3}{2} \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{3 \sec^{2} (t)}{2} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{3 \sec^{2} (t)}{2}$ is positive.

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{4 {(\frac{3}{2} \tan (t))}^{2} + 9}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{4 {(\frac{3}{2} \tan (t))}^{2} + 9}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Combine $\frac{3}{2}$ and $\tan (t)$ .

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{4 {(\frac{3 \tan (t)}{2})}^{2} + 9}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.1.2

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

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Step 2.1.1.2.1

Apply the product rule to $\frac{3 \tan (t)}{2}$ .

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{4 \frac{{(3 \tan (t))}^{2}}{2^{2}} + 9}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.1.2.2

Apply the product rule to $3 \tan (t)$ .

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{4 \frac{3^{2} \tan^{2} (t)}{2^{2}} + 9}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.1.3

Raise $3$ to the power of $2$ .

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{4 \frac{9 \tan^{2} (t)}{2^{2}} + 9}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.1.4

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

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{4 \frac{9 \tan^{2} (t)}{4} + 9}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.1.5

Cancel the common factor of $4$ .

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Step 2.1.1.5.1

Cancel the common factor.

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{4 \frac{9 \tan^{2} (t)}{4} + 9}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.1.5.2

Rewrite the expression.

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{9 \tan^{2} (t) + 9}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.2

Factor $9$ out of $9 \tan^{2} (t) + 9$ .

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Step 2.1.2.1

Factor $9$ out of $9 \tan^{2} (t)$ .

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{9 (\tan^{2} (t)) + 9}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.2.2

Factor $9$ out of $9$ .

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{9 (\tan^{2} (t)) + 9 (1)}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.2.3

Factor $9$ out of $9 (\tan^{2} (t)) + 9 (1)$ .

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{9 (\tan^{2} (t) + 1)}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.3

Apply pythagorean identity.

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{9 \sec^{2} (t)}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.4

Rewrite $9 \sec^{2} (t)$ as ${(3 \sec (t))}^{2}$ .

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{\sqrt{{(3 \sec (t))}^{2}}} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.1.5

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

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3}{2} \tan (t))}^{3}}{3 \sec (t)} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2

Combine fractions.

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Step 2.2.1

Combine $\frac{3}{2}$ and $\tan (t)$ .

$\int_{0}^{\frac{π}{3}} \frac{{(\frac{3 \tan (t)}{2})}^{3}}{3 \sec (t)} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Apply the product rule to $\frac{3 \tan (t)}{2}$ .

$\int_{0}^{\frac{π}{3}} \frac{\frac{{(3 \tan (t))}^{3}}{2^{3}}}{3 \sec (t)} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2.2.2

Apply the product rule to $3 \tan (t)$ .

$\int_{0}^{\frac{π}{3}} \frac{\frac{3^{3} \tan^{3} (t)}{2^{3}}}{3 \sec (t)} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2.2.3

Simplify.

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Step 2.2.2.3.1

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

$\int_{0}^{\frac{π}{3}} \frac{\frac{27 \tan^{3} (t)}{2^{3}}}{3 \sec (t)} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2.2.3.2

Raise $2$ to the power of $3$ .

$\int_{0}^{\frac{π}{3}} \frac{\frac{27 \tan^{3} (t)}{8}}{3 \sec (t)} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2.2.3.3

Rewrite $\frac{\frac{27 \tan^{3} (t)}{8}}{3 \sec (t)}$ as a product.

$\int_{0}^{\frac{π}{3}} \frac{27 \tan^{3} (t)}{8} \cdot \frac{1}{3 \sec (t)} \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2.2.3.4

Multiply $\frac{27 \tan^{3} (t)}{8}$ by $\frac{1}{3 \sec (t)}$ .

$\int_{0}^{\frac{π}{3}} \frac{27 \tan^{3} (t)}{8 (3 \sec (t))} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2.2.3.5

Multiply $3$ by $8$ .

$\int_{0}^{\frac{π}{3}} \frac{27 \tan^{3} (t)}{24 \sec (t)} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2.2.3.6

Cancel the common factor of $27$ and $24$ .

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Step 2.2.2.3.6.1

Factor $3$ out of $27 \tan^{3} (t)$ .

$\int_{0}^{\frac{π}{3}} \frac{3 (9 \tan^{3} (t))}{24 \sec (t)} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2.2.3.6.2

Cancel the common factors.

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Step 2.2.2.3.6.2.1

Factor $3$ out of $24 \sec (t)$ .

$\int_{0}^{\frac{π}{3}} \frac{3 (9 \tan^{3} (t))}{3 (8 \sec (t))} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2.2.3.6.2.2

Cancel the common factor.

$\int_{0}^{\frac{π}{3}} \frac{3 (9 \tan^{3} (t))}{3 (8 \sec (t))} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2.2.3.6.2.3

Rewrite the expression.

$\int_{0}^{\frac{π}{3}} \frac{9 \tan^{3} (t)}{8 \sec (t)} \cdot \frac{3 \sec^{2} (t)}{2} d t$

Step 2.2.2.3.7

Multiply $\frac{9 \tan^{3} (t)}{8 \sec (t)}$ by $\frac{3 \sec^{2} (t)}{2}$ .

$\int_{0}^{\frac{π}{3}} \frac{9 \tan^{3} (t) (3 \sec^{2} (t))}{8 \sec (t) \cdot 2} d t$

Step 2.2.2.3.8

Multiply $3$ by $9$ .

$\int_{0}^{\frac{π}{3}} \frac{27 \tan^{3} (t) \sec^{2} (t)}{8 \sec (t) \cdot 2} d t$

Step 2.2.2.3.9

Multiply $2$ by $8$ .

$\int_{0}^{\frac{π}{3}} \frac{27 \tan^{3} (t) \sec^{2} (t)}{16 \sec (t)} d t$

Step 2.2.2.3.10

Cancel the common factor of $\sec^{2} (t)$ and $\sec (t)$ .

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Step 2.2.2.3.10.1

Factor $\sec (t)$ out of $27 \tan^{3} (t) \sec^{2} (t)$ .

$\int_{0}^{\frac{π}{3}} \frac{\sec (t) (27 \tan^{3} (t) \sec (t))}{16 \sec (t)} d t$

Step 2.2.2.3.10.2

Cancel the common factors.

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Step 2.2.2.3.10.2.1

Factor $\sec (t)$ out of $16 \sec (t)$ .

$\int_{0}^{\frac{π}{3}} \frac{\sec (t) (27 \tan^{3} (t) \sec (t))}{\sec (t) \cdot 16} d t$

Step 2.2.2.3.10.2.2

Cancel the common factor.

$\int_{0}^{\frac{π}{3}} \frac{\sec (t) (27 \tan^{3} (t) \sec (t))}{\sec (t) \cdot 16} d t$

Step 2.2.2.3.10.2.3

Rewrite the expression.

$\int_{0}^{\frac{π}{3}} \frac{27 \tan^{3} (t) \sec (t)}{16} d t$

Step 3

Since $\frac{27}{16}$ is constant with respect to $t$ , move $\frac{27}{16}$ out of the integral.

$\frac{27}{16} \int_{0}^{\frac{π}{3}} \tan^{3} (t) \sec (t) d t$

Step 4

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

$\frac{27}{16} \int_{0}^{\frac{π}{3}} \tan^{3} (t) \sec^{1} (t) d t$

Step 5

Factor out $\tan^{2} (t)$ .

$\frac{27}{16} \int_{0}^{\frac{π}{3}} \tan^{2} (t) \tan (t) \sec^{1} (t) d t$

Step 6

Using the Pythagorean Identity, rewrite $\tan^{2} (t)$ as $- 1 + \sec^{2} (t)$ .

$\frac{27}{16} \int_{0}^{\frac{π}{3}} (- 1 + \sec^{2} (t)) \tan (t) \sec^{1} (t) d t$

Step 7

Simplify.

$\frac{27}{16} \int_{0}^{\frac{π}{3}} (- 1 + \sec^{2} (t)) \tan (t) \sec (t) d t$

Step 8

Let $u = \sec (t)$ . Then $d u = \sec (t) \tan (t) d t$ , so $\frac{1}{\sec (t) \tan (t)} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = \sec (t)$ . Find $\frac{d u}{d t}$ .

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Step 8.1.1

Differentiate $\sec (t)$ .

$\frac{d}{d t} [\sec (t)]$

Step 8.1.2

The derivative of $\sec (t)$ with respect to $t$ is $\sec (t) \tan (t)$ .

$\sec (t) \tan (t)$

Step 8.2

Substitute the lower limit in for $t$ in $u = \sec (t)$ .

$u_{lower} = \sec (0)$

Step 8.3

The exact value of $\sec (0)$ is $1$ .

$u_{lower} = 1$

Step 8.4

Substitute the upper limit in for $t$ in $u = \sec (t)$ .

$u_{upper} = \sec (\frac{π}{3})$

Step 8.5

The exact value of $\sec (\frac{π}{3})$ is $2$ .

$u_{upper} = 2$

Step 8.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 2$

Step 8.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\frac{27}{16} \int_{1}^{2} - 1 + u^{2} d u$

Step 9

Split the single integral into multiple integrals.

$\frac{27}{16} (\int_{1}^{2} - 1 d u + \int_{1}^{2} u^{2} d u)$

Step 10

Apply the constant rule.

$\frac{27}{16} (- u]_{1}^{2} + \int_{1}^{2} u^{2} d u)$

Step 11

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{27}{16} (- u]_{1}^{2} + \frac{1}{3} u^{3}]_{1}^{2})$

Step 12

Combine $- u]_{1}^{2}$ and $\frac{1}{3} u^{3}]_{1}^{2}$ .

$\frac{27}{16} - u + \frac{1}{3} u^{3}]_{1}^{2}$

Step 13

Substitute and simplify.

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Step 13.1

Evaluate $- u + \frac{1}{3} u^{3}$ at $2$ and at $1$ .

$\frac{27}{16} ((- 1 \cdot 2 + \frac{1}{3} \cdot 2^{3}) - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3}))$

Step 13.2

Simplify.

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Step 13.2.1

Multiply $- 1$ by $2$ .

$\frac{27}{16} (- 2 + \frac{1}{3} \cdot 2^{3} - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3}))$

Step 13.2.2

Raise $2$ to the power of $3$ .

$\frac{27}{16} (- 2 + \frac{1}{3} \cdot 8 - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3}))$

Step 13.2.3

Combine $\frac{1}{3}$ and $8$ .

$\frac{27}{16} (- 2 + \frac{8}{3} - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3}))$

Step 13.2.4

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{27}{16} (- 2 \cdot \frac{3}{3} + \frac{8}{3} - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3}))$

Step 13.2.5

Combine $- 2$ and $\frac{3}{3}$ .

$\frac{27}{16} (\frac{- 2 \cdot 3}{3} + \frac{8}{3} - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3}))$

Step 13.2.6

Combine the numerators over the common denominator.

$\frac{27}{16} (\frac{- 2 \cdot 3 + 8}{3} - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3}))$

Step 13.2.7

Simplify the numerator.

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Step 13.2.7.1

Multiply $- 2$ by $3$ .

$\frac{27}{16} (\frac{- 6 + 8}{3} - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3}))$

Step 13.2.7.2

Add $- 6$ and $8$ .

$\frac{27}{16} (\frac{2}{3} - (- 1 \cdot 1 + \frac{1}{3} \cdot 1^{3}))$

Step 13.2.8

Multiply $- 1$ by $1$ .

$\frac{27}{16} (\frac{2}{3} - (- 1 + \frac{1}{3} \cdot 1^{3}))$

Step 13.2.9

One to any power is one.

$\frac{27}{16} (\frac{2}{3} - (- 1 + \frac{1}{3} \cdot 1))$

Step 13.2.10

Multiply $\frac{1}{3}$ by $1$ .

$\frac{27}{16} (\frac{2}{3} - (- 1 + \frac{1}{3}))$

Step 13.2.11

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{27}{16} (\frac{2}{3} - (- 1 \cdot \frac{3}{3} + \frac{1}{3}))$

Step 13.2.12

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{27}{16} (\frac{2}{3} - (\frac{- 1 \cdot 3}{3} + \frac{1}{3}))$

Step 13.2.13

Combine the numerators over the common denominator.

$\frac{27}{16} (\frac{2}{3} - \frac{- 1 \cdot 3 + 1}{3})$

Step 13.2.14

Simplify the numerator.

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Step 13.2.14.1

Multiply $- 1$ by $3$ .

$\frac{27}{16} (\frac{2}{3} - \frac{- 3 + 1}{3})$

Step 13.2.14.2

Add $- 3$ and $1$ .

$\frac{27}{16} (\frac{2}{3} - \frac{- 2}{3})$

Step 13.2.15

Move the negative in front of the fraction.

$\frac{27}{16} (\frac{2}{3} - - \frac{2}{3})$

Step 13.2.16

Multiply $- 1$ by $- 1$ .

$\frac{27}{16} (\frac{2}{3} + 1 (\frac{2}{3}))$

Step 13.2.17

Multiply $\frac{2}{3}$ by $1$ .

$\frac{27}{16} (\frac{2}{3} + \frac{2}{3})$

Step 13.2.18

Combine the numerators over the common denominator.

$\frac{27}{16} \cdot \frac{2 + 2}{3}$

Step 13.2.19

Add $2$ and $2$ .

$\frac{27}{16} \cdot \frac{4}{3}$

Step 13.2.20

Multiply $\frac{27}{16}$ by $\frac{4}{3}$ .

$\frac{27 \cdot 4}{16 \cdot 3}$

Step 13.2.21

Multiply $27$ by $4$ .

$\frac{108}{16 \cdot 3}$

Step 13.2.22

Multiply $16$ by $3$ .

$\frac{108}{48}$

Step 13.2.23

Cancel the common factor of $108$ and $48$ .

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Step 13.2.23.1

Factor $12$ out of $108$ .

$\frac{12 (9)}{48}$

Step 13.2.23.2

Cancel the common factors.

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Step 13.2.23.2.1

Factor $12$ out of $48$ .

$\frac{12 \cdot 9}{12 \cdot 4}$

Step 13.2.23.2.2

Cancel the common factor.

$\frac{12 \cdot 9}{12 \cdot 4}$

Step 13.2.23.2.3

Rewrite the expression.

$\frac{9}{4}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{4}$

Decimal Form:

$2.25$

Mixed Number Form:

$2 \frac{1}{4}$

Step 15