Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Let $u = \frac{1}{θ}$ . Find $\frac{d u}{d θ}$ .

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Step 2.1.1

Differentiate $\frac{1}{θ}$ .

$\frac{d}{d θ} [\frac{1}{θ}]$

Step 2.1.2

Rewrite $\frac{1}{θ}$ as $θ^{- 1}$ .

$\frac{d}{d θ} [θ^{- 1}]$

Step 2.1.3

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

$- θ^{- 2}$

Step 2.1.4

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

$- \frac{1}{θ^{2}}$

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

$- \sin (u) + C$

Step 6

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

$- \sin (\frac{1}{θ}) + C$