Calculus Examples

$\int 7 \cos (\sqrt{x}) d x$

Step 1

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int \cos (\sqrt{x}) d x$

Step 2

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 $d$ $u$ .

$7 \int 2 u \cos (u) d u$

Step 3

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

$7 (2 \int u \cos (u) d u)$

Step 4

Multiply $2$ by $7$ .

$14 \int u \cos (u) d u$

Step 5

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

$14 (u \sin (u) - \int \sin (u) d u)$

Step 6

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

$14 (u \sin (u) - (- \cos (u) + C))$

Step 7

Rewrite $14 (u \sin (u) - (- \cos (u) + C))$ as $14 (u \sin (u) + \cos (u)) + C$ .

$14 (u \sin (u) + \cos (u)) + C$

Step 8

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

$14 (\sqrt{x} \sin (\sqrt{x}) + \cos (\sqrt{x})) + C$