Calculus Examples

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

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{2 x}$ .

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

Combine $x$ and $\frac{e^{2 x}}{2}$ .

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

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

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

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Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Rewrite.

$\frac{2}{1}$

Divide $2$ by $1$ .

$2$

Rewrite the problem using $u$ and $d u$ .

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

Combine $e^{u}$ and $\frac{1}{2}$ .

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

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{x e^{2 x}}{2} - \frac{1}{2} (\frac{1}{2} \int e^{u} d u)$

Combine fractions.

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Multiply $\frac{1}{2}$ and $\frac{1}{2}$ .

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

Multiply $2$ by $2$ .

$\frac{x e^{2 x}}{2} - \frac{1}{4} \int e^{u} d u$

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

$\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u} + C)$

Rewrite $\frac{x e^{2 x}}{2} - \frac{1}{4} (e^{u} + C)$ as $\frac{x e^{2 x}}{2} - \frac{e^{u}}{4} + C$ .

$\frac{x e^{2 x}}{2} - \frac{e^{u}}{4} + C$

Replace all occurrences of $u$ with $2 x$ .

$\frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4} + C$

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