Calculus Examples

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

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 $d$ $u$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $- 1$ .

$2 (u (- \cos (u)) + 1 \int \cos (u) d u)$

Step 5.2

Multiply $\int \cos (u) d u$ by $1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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