Calculus Examples

$\int_{0}^{1} \frac{r^{3}}{\sqrt{4 + r^{2}}} d r$

Step 1

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

$\int_{0}^{0.4636476} \frac{{(2 \tan (t))}^{3}}{\sqrt{4 + {(2 \tan (t))}^{2}}} (2 \sec^{2} (t)) d t$

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the product rule to $2 \tan (t)$ .

$\int_{0}^{0.4636476} \frac{{(2 \tan (t))}^{3}}{\sqrt{4 + 2^{2} \tan^{2} (t)}} (2 \sec^{2} (t)) d t$

Step 2.1.1.2

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

$\int_{0}^{0.4636476} \frac{{(2 \tan (t))}^{3}}{\sqrt{4 + 4 \tan^{2} (t)}} (2 \sec^{2} (t)) d t$

Step 2.1.2

Factor $4$ out of $4$ .

$\int_{0}^{0.4636476} \frac{{(2 \tan (t))}^{3}}{\sqrt{4 (1) + 4 \tan^{2} (t)}} (2 \sec^{2} (t)) d t$

Step 2.1.3

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

$\int_{0}^{0.4636476} \frac{{(2 \tan (t))}^{3}}{\sqrt{4 (1) + 4 (\tan^{2} (t))}} (2 \sec^{2} (t)) d t$

Step 2.1.4

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

$\int_{0}^{0.4636476} \frac{{(2 \tan (t))}^{3}}{\sqrt{4 (1 + \tan^{2} (t))}} (2 \sec^{2} (t)) d t$

Step 2.1.5

Rearrange terms.

$\int_{0}^{0.4636476} \frac{{(2 \tan (t))}^{3}}{\sqrt{4 (\tan^{2} (t) + 1)}} (2 \sec^{2} (t)) d t$

Step 2.1.6

Apply pythagorean identity.

$\int_{0}^{0.4636476} \frac{{(2 \tan (t))}^{3}}{\sqrt{4 \sec^{2} (t)}} (2 \sec^{2} (t)) d t$

Step 2.1.7

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

$\int_{0}^{0.4636476} \frac{{(2 \tan (t))}^{3}}{\sqrt{{(2 \sec (t))}^{2}}} (2 \sec^{2} (t)) d t$

Step 2.1.8

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

$\int_{0}^{0.4636476} \frac{{(2 \tan (t))}^{3}}{2 \sec (t)} (2 \sec^{2} (t)) d t$

Step 2.2

Reduce the expression by cancelling the common factors.

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Step 2.2.1

Cancel the common factor of $2 \sec (t)$ .

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Step 2.2.1.1

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

$\int_{0}^{0.4636476} \frac{{(2 \tan (t))}^{3}}{2 \sec (t)} (2 \sec (t) (\sec (t))) d t$

Step 2.2.1.2

Cancel the common factor.

$\int_{0}^{0.4636476} \frac{{(2 \tan (t))}^{3}}{2 \sec (t)} (2 \sec (t) \sec (t)) d t$

Step 2.2.1.3

Rewrite the expression.

$\int_{0}^{0.4636476} {(2 \tan (t))}^{3} \sec (t) d t$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Factor $2$ out of $2 \tan (t)$ .

$\int_{0}^{0.4636476} {(2 (\tan (t)))}^{3} \sec (t) d t$

Step 2.2.2.2

Apply the product rule to $2 (\tan (t))$ .

$\int_{0}^{0.4636476} 2^{3} \tan^{3} (t) \sec (t) d t$

Step 2.2.2.3

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

$\int_{0}^{0.4636476} 8 \tan^{3} (t) \sec (t) d t$

Step 3

Since $8$ is constant with respect to $t$ , move $8$ out of the integral.

$8 \int_{0}^{0.4636476} \tan^{3} (t) \sec (t) d t$

Step 4

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

$8 \int_{0}^{0.4636476} \tan^{3} (t) \sec^{1} (t) d t$

Step 5

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

$8 \int_{0}^{0.4636476} \tan^{2} (t) \tan (t) \sec^{1} (t) d t$

Step 6

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

$8 \int_{0}^{0.4636476} (- 1 + \sec^{2} (t)) \tan (t) \sec^{1} (t) d t$

Step 7

Simplify.

$8 \int_{0}^{0.4636476} (- 1 + \sec^{2} (t)) \tan (t) \sec (t) d t$

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 $u = \sec (t)$ .

$u_{upper} = \sec (0.4636476)$

Step 8.5

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

$u_{lower} = 1$

$u_{upper} = \sec (0.4636476)$

Step 8.6

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

$8 \int_{1}^{\sec (0.4636476)} - 1 + u^{2} d u$

Step 9

Split the single integral into multiple integrals.

$8 (\int_{1}^{\sec (0.4636476)} - 1 d u + \int_{1}^{\sec (0.4636476)} u^{2} d u)$

Step 10

Apply the constant rule.

$8 (- u]_{1}^{\sec (0.4636476)} + \int_{1}^{\sec (0.4636476)} u^{2} d u)$

Step 11

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

$8 (- u]_{1}^{\sec (0.4636476)} + \frac{1}{3} u^{3}]_{1}^{\sec (0.4636476)})$

Step 12

Simplify.

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Step 12.1

Combine $- u]_{1}^{\sec (0.4636476)}$ and $\frac{1}{3} u^{3}]_{1}^{\sec (0.4636476)}$ .

$8 (- u + \frac{1}{3} u^{3}]_{1}^{\sec (0.4636476)})$

Step 12.2

Combine $\frac{1}{3}$ and $u^{3}$ .

$8 (- u + \frac{u^{3}}{3}]_{1}^{\sec (0.4636476)})$

Step 13

Substitute and simplify.

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Step 13.1

Evaluate $- u + \frac{u^{3}}{3}$ at $\sec (0.4636476)$ and at $1$ .

$8 ((- \sec (0.4636476) + \frac{\sec^{3} (0.4636476)}{3}) - (- 1 \cdot 1 + \frac{1^{3}}{3}))$

Step 13.2

Simplify.

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Step 13.2.1

To write $- \sec (0.4636476)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$8 (- \sec (0.4636476) \cdot \frac{3}{3} + \frac{\sec^{3} (0.4636476)}{3} - (- 1 \cdot 1 + \frac{1^{3}}{3}))$

Step 13.2.2

Combine $- \sec (0.4636476)$ and $\frac{3}{3}$ .

$8 (\frac{- \sec (0.4636476) \cdot 3}{3} + \frac{\sec^{3} (0.4636476)}{3} - (- 1 \cdot 1 + \frac{1^{3}}{3}))$

Step 13.2.3

Combine the numerators over the common denominator.

$8 (\frac{- \sec (0.4636476) \cdot 3 + \sec^{3} (0.4636476)}{3} - (- 1 \cdot 1 + \frac{1^{3}}{3}))$

Step 13.2.4

Multiply $3$ by $- 1$ .

$8 (\frac{- 3 \sec (0.4636476) + \sec^{3} (0.4636476)}{3} - (- 1 \cdot 1 + \frac{1^{3}}{3}))$

Step 13.2.5

Multiply $- 1$ by $1$ .

$8 (\frac{- 3 \sec (0.4636476) + \sec^{3} (0.4636476)}{3} - (- 1 + \frac{1^{3}}{3}))$

Step 13.2.6

One to any power is one.

$8 (\frac{- 3 \sec (0.4636476) + \sec^{3} (0.4636476)}{3} - (- 1 + \frac{1}{3}))$

Step 13.2.7

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

$8 (\frac{- 3 \sec (0.4636476) + \sec^{3} (0.4636476)}{3} - (- 1 \cdot \frac{3}{3} + \frac{1}{3}))$

Step 13.2.8

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

$8 (\frac{- 3 \sec (0.4636476) + \sec^{3} (0.4636476)}{3} - (\frac{- 1 \cdot 3}{3} + \frac{1}{3}))$

Step 13.2.9

Combine the numerators over the common denominator.

$8 (\frac{- 3 \sec (0.4636476) + \sec^{3} (0.4636476)}{3} - \frac{- 1 \cdot 3 + 1}{3})$

Step 13.2.10

Simplify the numerator.

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Step 13.2.10.1

Multiply $- 1$ by $3$ .

$8 (\frac{- 3 \sec (0.4636476) + \sec^{3} (0.4636476)}{3} - \frac{- 3 + 1}{3})$

Step 13.2.10.2

Add $- 3$ and $1$ .

$8 (\frac{- 3 \sec (0.4636476) + \sec^{3} (0.4636476)}{3} - \frac{- 2}{3})$

Step 13.2.11

Move the negative in front of the fraction.

$8 (\frac{- 3 \sec (0.4636476) + \sec^{3} (0.4636476)}{3} - - \frac{2}{3})$

Step 13.2.12

Multiply $- 1$ by $- 1$ .

$8 (\frac{- 3 \sec (0.4636476) + \sec^{3} (0.4636476)}{3} + 1 (\frac{2}{3}))$

Step 13.2.13

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

$8 (\frac{- 3 \sec (0.4636476) + \sec^{3} (0.4636476)}{3} + \frac{2}{3})$

Step 14

Simplify.

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Step 14.1

Factor $- 1$ out of $- 3 \sec (0.4636476)$ .

$8 (\frac{- (3 \sec (0.4636476)) + \sec^{3} (0.4636476)}{3} + \frac{2}{3})$

Step 14.2

Factor $- 1$ out of $\sec^{3} (0.4636476)$ .

$8 (\frac{- (3 \sec (0.4636476)) - 1 (- \sec^{3} (0.4636476))}{3} + \frac{2}{3})$

Step 14.3

Factor $- 1$ out of $- (3 \sec (0.4636476)) - 1 (- \sec^{3} (0.4636476))$ .

$8 (\frac{- (3 \sec (0.4636476) - \sec^{3} (0.4636476))}{3} + \frac{2}{3})$

Step 14.4

Rewrite $- (3 \sec (0.4636476) - \sec^{3} (0.4636476))$ as $- 1 (3 \sec (0.4636476) - \sec^{3} (0.4636476))$ .

$8 (\frac{- 1 (3 \sec (0.4636476) - \sec^{3} (0.4636476))}{3} + \frac{2}{3})$

Step 14.5

Move the negative in front of the fraction.

$8 (- \frac{3 \sec (0.4636476) - \sec^{3} (0.4636476)}{3} + \frac{2}{3})$

Step 15

The result can be shown in multiple forms.

Exact Form:

$8 (- \frac{3 \sec (0.4636476) - \sec^{3} (0.4636476)}{3} + \frac{2}{3})$

Decimal Form:

$0.11584138 \dots$

Step 16