Calculus Examples

$\int \frac{\cos (\ln (x))}{x} d x$

Step 1

Let $u = \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = \ln (x)$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $\ln (x)$ .

$\frac{d}{d x} [\ln (x)]$

Step 1.1.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{1}{x}$

Step 1.2

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

$\int \cos (u) d u$

Step 2

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

$\sin (u) + C$

Step 3

Replace all occurrences of $u$ with $\ln (x)$ .

$\sin (\ln (x)) + C$