Calculus Examples

$\int \frac{1}{θ^{2}} \sin (\frac{1}{θ}) \cos (\frac{1}{θ}) d θ$

Step 1

Simplify.

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Step 1.1

Combine $\sin (\frac{1}{θ})$ and $\frac{1}{θ^{2}}$ .

$\int \frac{\sin (\frac{1}{θ})}{θ^{2}} \cdot \cos (\frac{1}{θ}) d θ$

Step 1.2

Combine $\frac{\sin (\frac{1}{θ})}{θ^{2}}$ and $\cos (\frac{1}{θ})$ .

$\int \frac{\sin (\frac{1}{θ}) \cos (\frac{1}{θ})}{θ^{2}} d θ$

Step 2

Let $u_{2} = \sin (\frac{1}{θ})$ . Then $d u_{2} = - \frac{\cos (\frac{1}{θ})}{θ^{2}} d θ$ , so $- d u_{2} = \frac{\cos (\frac{1}{θ})}{θ^{2}} d θ$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.1

Let $u_{2} = \sin (\frac{1}{θ})$ . Find $\frac{d u_{2}}{d θ}$ .

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Step 2.1.1

Differentiate $\sin (\frac{1}{θ})$ .

$\frac{d}{d θ} [\sin (\frac{1}{θ})]$

Step 2.1.2

Differentiate using the chain rule, which states that $\frac{d}{d θ} [f (g (θ))]$ is $f' (g (θ)) g' (θ)$ where $f (θ) = \sin (θ)$ and $g (θ) = \frac{1}{θ}$ .

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Step 2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $\frac{1}{θ}$ .

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d θ} [\frac{1}{θ}]$

Step 2.1.2.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$\cos (u_{1}) \frac{d}{d θ} [\frac{1}{θ}]$

Step 2.1.2.3

Replace all occurrences of $u_{1}$ with $\frac{1}{θ}$ .

$\cos (\frac{1}{θ}) \frac{d}{d θ} [\frac{1}{θ}]$

Step 2.1.3

Differentiate using the Power Rule.

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Step 2.1.3.1

Rewrite $\frac{1}{θ}$ as $θ^{- 1}$ .

$\cos (\frac{1}{θ}) \frac{d}{d θ} [θ^{- 1}]$

Step 2.1.3.2

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

$\cos (\frac{1}{θ}) (- θ^{- 2})$

Step 2.1.4

Simplify.

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Step 2.1.4.1

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

$\cos (\frac{1}{θ}) (- \frac{1}{θ^{2}})$

Step 2.1.4.2

Combine $\cos (\frac{1}{θ})$ and $\frac{1}{θ^{2}}$ .

$- \frac{\cos (\frac{1}{θ})}{θ^{2}}$

Step 2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int - u_{2} d u_{2}$

Step 3

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$- \int u_{2} d u_{2}$

Step 4

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

$- (\frac{1}{2} {u_{2}}^{2} + C)$

Step 5

Rewrite $- (\frac{1}{2} {u_{2}}^{2} + C)$ as $- \frac{1}{2} {u_{2}}^{2} + C$ .

$- \frac{1}{2} {u_{2}}^{2} + C$

Step 6

Replace all occurrences of $u_{2}$ with $\sin (\frac{1}{θ})$ .

$- \frac{1}{2} \sin^{2} (\frac{1}{θ}) + C$