Calculus Examples

$16 x + \frac{5 x}{\sqrt{9 - x^{2}}}$

Step 1

Write $16 x + \frac{5 x}{\sqrt{9 - x^{2}}}$ as a function.

$f (x) = 16 x + \frac{5 x}{\sqrt{9 - x^{2}}}$

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.

$F (x) = \int 16 x + \frac{5 x}{\sqrt{9 - x^{2}}} d x$

Step 4

Split the single integral into multiple integrals.

$\int 16 x d x + \int \frac{5 x}{\sqrt{9 - x^{2}}} d x$

Step 5

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$16 \int x d x + \int \frac{5 x}{\sqrt{9 - x^{2}}} d x$

Step 6

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

$16 (\frac{1}{2} x^{2} + C) + \int \frac{5 x}{\sqrt{9 - x^{2}}} d x$

Step 7

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$16 (\frac{1}{2} x^{2} + C) + 5 \int \frac{x}{\sqrt{9 - x^{2}}} d x$

Step 8

Let $u = 9 - x^{2}$ . Then $d u = - 2 x d x$ , so $- \frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = 9 - x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $9 - x^{2}$ .

$\frac{d}{d x} [9 - x^{2}]$

Step 8.1.2

Differentiate.

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Step 8.1.2.1

By the Sum Rule, the derivative of $9 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$

Step 8.1.2.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x^{2}]$

Step 8.1.3

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 8.1.3.1

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

$0 - \frac{d}{d x} [x^{2}]$

Step 8.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - (2 x)$

Step 8.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 8.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 8.2

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

$16 (\frac{1}{2} x^{2} + C) + 5 \int \frac{1}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 9

Simplify.

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Step 9.1

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

$16 (\frac{x^{2}}{2} + C) + 5 \int \frac{1}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 9.2

Move the negative in front of the fraction.

$16 (\frac{x^{2}}{2} + C) + 5 \int \frac{1}{\sqrt{u}} (- \frac{1}{2}) d u$

Step 9.3

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

$16 (\frac{x^{2}}{2} + C) + 5 \int - \frac{1}{\sqrt{u} \cdot 2} d u$

Step 9.4

Move $2$ to the left of $\sqrt{u}$ .

$16 (\frac{x^{2}}{2} + C) + 5 \int - \frac{1}{2 \sqrt{u}} d u$

Step 10

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

$16 (\frac{x^{2}}{2} + C) + 5 (- \int \frac{1}{2 \sqrt{u}} d u)$

Step 11

Multiply $- 1$ by $5$ .

$16 (\frac{x^{2}}{2} + C) - 5 \int \frac{1}{2 \sqrt{u}} d u$

Step 12

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

$16 (\frac{x^{2}}{2} + C) - 5 (\frac{1}{2} \int \frac{1}{\sqrt{u}} d u)$

Step 13

Simplify the expression.

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Step 13.1

Simplify.

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Step 13.1.1

Combine $\frac{1}{2}$ and $- 5$ .

$16 (\frac{x^{2}}{2} + C) + \frac{- 5}{2} \int \frac{1}{\sqrt{u}} d u$

Step 13.1.2

Move the negative in front of the fraction.

$16 (\frac{x^{2}}{2} + C) - \frac{5}{2} \int \frac{1}{\sqrt{u}} d u$

Step 13.2

Apply basic rules of exponents.

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Step 13.2.1

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

$16 (\frac{x^{2}}{2} + C) - \frac{5}{2} \int \frac{1}{u^{\frac{1}{2}}} d u$

Step 13.2.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$16 (\frac{x^{2}}{2} + C) - \frac{5}{2} \int {(u^{\frac{1}{2}})}^{- 1} d u$

Step 13.2.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 13.2.3.1

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

$16 (\frac{x^{2}}{2} + C) - \frac{5}{2} \int u^{\frac{1}{2} \cdot - 1} d u$

Step 13.2.3.2

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

$16 (\frac{x^{2}}{2} + C) - \frac{5}{2} \int u^{\frac{- 1}{2}} d u$

Step 13.2.3.3

Move the negative in front of the fraction.

$16 (\frac{x^{2}}{2} + C) - \frac{5}{2} \int u^{- \frac{1}{2}} d u$

Step 14

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

$16 (\frac{x^{2}}{2} + C) - \frac{5}{2} (2 u^{\frac{1}{2}} + C)$

Step 15

Simplify.

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Step 15.1

Simplify.

$8 x^{2} - \frac{5}{2} (2 u^{\frac{1}{2}}) + C$

Step 15.2

Simplify.

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Step 15.2.1

Multiply $2$ by $- 1$ .

$8 x^{2} - 2 (\frac{5}{2}) u^{\frac{1}{2}} + C$

Step 15.2.2

Combine $- 2$ and $\frac{5}{2}$ .

$8 x^{2} + \frac{- 2 \cdot 5}{2} u^{\frac{1}{2}} + C$

Step 15.2.3

Multiply $- 2$ by $5$ .

$8 x^{2} + \frac{- 10}{2} u^{\frac{1}{2}} + C$

Step 15.2.4

Combine $\frac{- 10}{2}$ and $u^{\frac{1}{2}}$ .

$8 x^{2} + \frac{- 10 u^{\frac{1}{2}}}{2} + C$

Step 15.2.5

Factor $2$ out of $- 10 u^{\frac{1}{2}}$ .

$8 x^{2} + \frac{2 (- 5 u^{\frac{1}{2}})}{2} + C$

Step 15.2.6

Cancel the common factors.

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Step 15.2.6.1

Factor $2$ out of $2$ .

$8 x^{2} + \frac{2 (- 5 u^{\frac{1}{2}})}{2 (1)} + C$

Step 15.2.6.2

Cancel the common factor.

$8 x^{2} + \frac{2 (- 5 u^{\frac{1}{2}})}{2 \cdot 1} + C$

Step 15.2.6.3

Rewrite the expression.

$8 x^{2} + \frac{- 5 u^{\frac{1}{2}}}{1} + C$

Step 15.2.6.4

Divide $- 5 u^{\frac{1}{2}}$ by $1$ .

$8 x^{2} - 5 u^{\frac{1}{2}} + C$

Step 16

Replace all occurrences of $u$ with $9 - x^{2}$ .

$8 x^{2} - 5 {(9 - x^{2})}^{\frac{1}{2}} + C$

Step 17

The answer is the antiderivative of the function $f (x) = 16 x + \frac{5 x}{\sqrt{9 - x^{2}}}$ .

$F (x) =$ $8 x^{2} - 5 {(9 - x^{2})}^{\frac{1}{2}} + C$