Calculus Examples

$\int \sin (x) \ln (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = \sin (x)$ .

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

Step 2

Combine $\frac{1}{x}$ and $\cos (x)$ .

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$\ln (x) (- \cos (x)) + 1 \int \frac{\cos (x)}{x} d x$

Step 4.2

Multiply $\int \frac{\cos (x)}{x} d x$ by $1$ .

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

Step 5

$\int \frac{\cos (x)}{x} d x$ is a special integral. The integral is the cosine integral function.

$\ln (x) (- \cos (x)) + C i (x) + C$

Step 6

Rewrite $\ln (x) (- \cos (x)) + C i (x) + C$ as $- \ln (x) \cos (x) + C i (x) + C$ .

$- \ln (x) \cos (x) + C i (x) + C$