Calculus Examples

$\int e^{\sqrt{x}} d x$

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 $d$ $u$ .

$\int 2 u e^{u} d u$

Step 2

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

$2 \int u e^{u} d u$

Step 3

Integrate by parts using the formula $\int w d v = w v - \int v d w$ , where $w = u$ and $dv = e^{u}$ .

$2 (u e^{u} - \int e^{u} d u)$

Step 4

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$2 (u e^{u} - (e^{u} + C))$

Step 5

Simplify.

$2 (u e^{u} - e^{u}) + C$

Step 6

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

$2 (\sqrt{x} e^{\sqrt{x}} - e^{\sqrt{x}}) + C$