Calculus Examples

$\int e^{x^{\frac{1}{2}}} d x$

Step 1

Let $u = x^{\frac{1}{2}}$ . 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 $x^{\frac{1}{2}}$ .

$2 (x^{\frac{1}{2}} e^{x^{\frac{1}{2}}} - e^{x^{\frac{1}{2}}}) + C$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$2 (x^{\frac{1}{2}} e^{x^{\frac{1}{2}}}) + 2 (- e^{x^{\frac{1}{2}}}) + C$

Step 7.2

Multiply $- 1$ by $2$ .

$2 x^{\frac{1}{2}} e^{x^{\frac{1}{2}}} - 2 e^{x^{\frac{1}{2}}} + C$