Calculus Examples

$\int \frac{9}{θ^{2}} \cos (\frac{9}{θ}) d θ$

Step 1

Combine $\frac{9}{θ^{2}}$ and $\cos (\frac{9}{θ})$ .

$\int \frac{9 \cos (\frac{9}{θ})}{θ^{2}} d θ$

Step 2

Since $9$ is constant with respect to $θ$ , move $9$ out of the integral.

$9 \int \frac{\cos (\frac{9}{θ})}{θ^{2}} d θ$

Step 3

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

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Step 3.1

Let $u = \frac{9}{θ}$ . Find $\frac{d u}{d θ}$ .

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Step 3.1.1

Differentiate $\frac{9}{θ}$ .

$\frac{d}{d θ} [\frac{9}{θ}]$

Step 3.1.2

Since $9$ is constant with respect to $θ$ , the derivative of $\frac{9}{θ}$ with respect to $θ$ is $9 \frac{d}{d θ} [\frac{1}{θ}]$ .

$9 \frac{d}{d θ} [\frac{1}{θ}]$

Step 3.1.3

Rewrite $\frac{1}{θ}$ as $θ^{- 1}$ .

$9 \frac{d}{d θ} [θ^{- 1}]$

Step 3.1.4

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

$9 (- θ^{- 2})$

Step 3.1.5

Multiply $- 1$ by $9$ .

$- 9 θ^{- 2}$

Step 3.1.6

Simplify.

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Step 3.1.6.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 9 \frac{1}{θ^{2}}$

Step 3.1.6.2

Combine terms.

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Step 3.1.6.2.1

Combine $- 9$ and $\frac{1}{θ^{2}}$ .

$\frac{- 9}{θ^{2}}$

Step 3.1.6.2.2

Move the negative in front of the fraction.

$- \frac{9}{θ^{2}}$

Step 3.2

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

$9 \int - \frac{1}{9} \cos (u) d u$

Step 4

Since $- \frac{1}{9}$ is constant with respect to $u$ , move $- \frac{1}{9}$ out of the integral.

$9 (- \frac{1}{9} \int \cos (u) d u)$

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $9$ .

$- 9 (\frac{1}{9} \int \cos (u) d u)$

Step 5.2

Combine $\frac{1}{9}$ and $- 9$ .

$\frac{- 9}{9} \int \cos (u) d u$

Step 5.3

Cancel the common factor of $- 9$ and $9$ .

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Step 5.3.1

Factor $9$ out of $- 9$ .

$\frac{9 \cdot - 1}{9} \int \cos (u) d u$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $9$ out of $9$ .

$\frac{9 \cdot - 1}{9 (1)} \int \cos (u) d u$

Step 5.3.2.2

Cancel the common factor.

$\frac{9 \cdot - 1}{9 \cdot 1} \int \cos (u) d u$

Step 5.3.2.3

Rewrite the expression.

$\frac{- 1}{1} \int \cos (u) d u$

Step 5.3.2.4

Divide $- 1$ by $1$ .

$- \int \cos (u) d u$

Step 6

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

$- (\sin (u) + C)$

Step 7

Simplify.

$- \sin (u) + C$

Step 8

Replace all occurrences of $u$ with $\frac{9}{θ}$ .

$- \sin (\frac{9}{θ}) + C$