Calculus Examples

$\int \cos (\sqrt{x}) d x$

Step 1

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

$\int 2 u \cos (u) d u$

Step 2

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$2 \int u \cos (u) d u$

Step 3

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u$ and $dv = \cos (u)$ .

$2 (u \sin (u) - \int \sin (u) d u)$

Step 4

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

$2 (u \sin (u) - (- \cos (u) + C))$

Step 5

Rewrite $2 (u \sin (u) - (- \cos (u) + C))$ as $2 (u \sin (u) + \cos (u)) + C$ .

$2 (u \sin (u) + \cos (u)) + C$

Step 6

Replace all occurrences of $u$ with $\sqrt{x}$ .

$2 (\sqrt{x} \sin (\sqrt{x}) + \cos (\sqrt{x})) + C$