Calculus Examples

$x^{3} \sqrt{x^{2} + 9}$

Step 1

Write $x^{3} \sqrt{x^{2} + 9}$ as a function.

$f (x) = x^{3} \sqrt{x^{2} + 9}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x^{3} \sqrt{x^{2} + 9} d x$

Step 4

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

$\int {(3 \tan (t))}^{3} \sqrt{{(3 \tan (t))}^{2} + 9} (3 \sec^{2} (t)) d t$

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

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

$\int {(3 \tan (t))}^{3} \sqrt{3^{2} \tan^{2} (t) + 9} (3 \sec^{2} (t)) d t$

Step 5.1.1.2

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

$\int {(3 \tan (t))}^{3} \sqrt{9 \tan^{2} (t) + 9} (3 \sec^{2} (t)) d t$

Step 5.1.2

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

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Step 5.1.2.1

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

$\int {(3 \tan (t))}^{3} \sqrt{9 (\tan^{2} (t)) + 9} (3 \sec^{2} (t)) d t$

Step 5.1.2.2

Factor $9$ out of $9$ .

$\int {(3 \tan (t))}^{3} \sqrt{9 (\tan^{2} (t)) + 9 (1)} (3 \sec^{2} (t)) d t$

Step 5.1.2.3

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

$\int {(3 \tan (t))}^{3} \sqrt{9 (\tan^{2} (t) + 1)} (3 \sec^{2} (t)) d t$

Step 5.1.3

Apply pythagorean identity.

$\int {(3 \tan (t))}^{3} \sqrt{9 \sec^{2} (t)} (3 \sec^{2} (t)) d t$

Step 5.1.4

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

$\int {(3 \tan (t))}^{3} \sqrt{{(3 \sec (t))}^{2}} (3 \sec^{2} (t)) d t$

Step 5.1.5

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

$\int {(3 \tan (t))}^{3} (3 \sec (t)) (3 \sec^{2} (t)) d t$

Step 5.2

Simplify.

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Step 5.2.1

Factor $3$ out of $3 \tan (t)$ .

$\int {(3 (\tan (t)))}^{3} (3 \sec (t)) (3 \sec^{2} (t)) d t$

Step 5.2.2

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

$\int 3^{3} \tan^{3} (t) (3 \sec (t)) (3 \sec^{2} (t)) d t$

Step 5.2.3

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

$\int 27 \tan^{3} (t) (3 \sec (t)) (3 \sec^{2} (t)) d t$

Step 5.2.4

Multiply $3$ by $27$ .

$\int 81 \tan^{3} (t) \sec (t) (3 \sec^{2} (t)) d t$

Step 5.2.5

Multiply $3$ by $81$ .

$\int 243 \tan^{3} (t) \sec (t) \sec^{2} (t) d t$

Step 5.2.6

Multiply $\sec (t)$ by $\sec^{2} (t)$ by adding the exponents.

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Step 5.2.6.1

Move $\sec^{2} (t)$ .

$\int 243 \tan^{3} (t) (\sec^{2} (t) \sec (t)) d t$

Step 5.2.6.2

Multiply $\sec^{2} (t)$ by $\sec (t)$ .

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Step 5.2.6.2.1

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

$\int 243 \tan^{3} (t) (\sec^{2} (t) \sec^{1} (t)) d t$

Step 5.2.6.2.2

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

$\int 243 \tan^{3} (t) {\sec (t)}^{2 + 1} d t$

Step 5.2.6.3

Add $2$ and $1$ .

$\int 243 \tan^{3} (t) \sec^{3} (t) d t$

Step 6

Since $243$ is constant with respect to $t$ , move $243$ out of the integral.

$243 \int \tan^{3} (t) \sec^{3} (t) d t$

Step 7

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

$243 \int \tan^{2} (t) \tan (t) \sec^{3} (t) d t$

Step 8

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

$243 \int (- 1 + \sec^{2} (t)) \tan (t) \sec^{3} (t) d t$

Step 9

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 9.1

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

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Step 9.1.1

Differentiate $\sec (t)$ .

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

Step 9.1.2

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

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

Step 9.2

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

$243 \int (- 1 + u^{2}) u^{2} d u$

Step 10

Multiply $(- 1 + u^{2}) u^{2}$ .

$243 \int - 1 u^{2} + u^{2} u^{2} d u$

Step 11

Simplify.

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Step 11.1

Rewrite $- 1 u^{2}$ as $- u^{2}$ .

$243 \int - u^{2} + u^{2} u^{2} d u$

Step 11.2

Multiply $u^{2}$ by $u^{2}$ by adding the exponents.

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Step 11.2.1

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

$243 \int - u^{2} + u^{2 + 2} d u$

Step 11.2.2

Add $2$ and $2$ .

$243 \int - u^{2} + u^{4} d u$

Step 12

Split the single integral into multiple integrals.

$243 (\int - u^{2} d u + \int u^{4} d u)$

Step 13

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$243 (- \int u^{2} d u + \int u^{4} d u)$

Step 14

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

$243 (- (\frac{1}{3} u^{3} + C) + \int u^{4} d u)$

Step 15

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$243 (- (\frac{1}{3} u^{3} + C) + \frac{1}{5} u^{5} + C)$

Step 16

Simplify.

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Step 16.1

Simplify.

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Step 16.1.1

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

$243 (- (\frac{u^{3}}{3} + C) + \frac{1}{5} u^{5} + C)$

Step 16.1.2

Combine $\frac{1}{5}$ and $u^{5}$ .

$243 (- (\frac{u^{3}}{3} + C) + \frac{u^{5}}{5} + C)$

Step 16.2

Simplify.

$243 (- \frac{1}{3} u^{3} + \frac{1}{5} u^{5}) + C$

Step 17

Substitute back in for each integration substitution variable.

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Step 17.1

Replace all occurrences of $u$ with $\sec (t)$ .

$243 (- \frac{1}{3} \sec^{3} (t) + \frac{1}{5} \sec^{5} (t)) + C$

Step 17.2

Replace all occurrences of $t$ with $\arctan (\frac{x}{3})$ .

$243 (- \frac{1}{3} \sec^{3} (\arctan (\frac{x}{3})) + \frac{1}{5} \sec^{5} (\arctan (\frac{x}{3}))) + C$

Step 18

Reorder terms.

$243 (- \frac{1}{3} \sec^{3} (\arctan (\frac{1}{3} x)) + \frac{1}{5} \sec^{5} (\arctan (\frac{1}{3} x))) + C$

Step 19

The answer is the antiderivative of the function $f (x) = x^{3} \sqrt{x^{2} + 9}$ .

$F (x) =$ $243 (- \frac{1}{3} \sec^{3} (\arctan (\frac{1}{3} x)) + \frac{1}{5} \sec^{5} (\arctan (\frac{1}{3} x))) + C$