Calculus Examples

\int x e^{4 x} d x

$\int x e^{4 x} d x$

Step 1

Integrate by parts using the formula

\int u d v = u v - \int v d u

$\int u d v = u v - \int v d u$ , where

u = x

$u = x$ and

d v = e^{4 x}

$d v = e^{4 x}$ .

x (\frac{1}{4} e^{4 x}) - \int \frac{1}{4} e^{4 x} d x

$x (\frac{1}{4} e^{4 x}) - \int \frac{1}{4} e^{4 x} d x$

Step 2

Simplify.

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Step 2.1

Combine

\frac{1}{4}

$\frac{1}{4}$ and

e^{4 x}

$e^{4 x}$ .

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

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

Step 2.2

Combine

x

$x$ and

\frac{e^{4 x}}{4}

$\frac{e^{4 x}}{4}$ .

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

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

Step 3

Since

$\frac{1}{4}$ is constant with respect to

$x$ , move

$\frac{1}{4}$ out of the integral.

$\frac{x e^{4 x}}{4} - (\frac{1}{4} \int e^{4 x} d x)$

Step 4

Let

$u = 4 x$ . Then

$d u = 4 d x$ , so

$\frac{1}{4} d u = d x$ . Rewrite using

$u$ and

$d$

$u$ .

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Step 4.1

Let

$u = 4 x$ . Find

$\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate

$4 x$ .

$\frac{d}{d x} [4 x]$

Step 4.1.2

Since

$4$ is constant with respect to

$x$ , the derivative of

$4 x$ with respect to

$x$ is

$4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 4.1.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$4 \cdot 1$

Step 4.1.4

Multiply

$4$ by

$1$ .

$4$

Step 4.2

Rewrite the problem using

$u$ and

$d u$ .

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

Step 5

Combine

$e^{u}$ and

$\frac{1}{4}$ .

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

Step 6

Since

$\frac{1}{4}$ is constant with respect to

$u$ , move

$\frac{1}{4}$ out of the integral.

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

Step 7

Simplify.

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Step 7.1

Multiply

$\frac{1}{4}$ by

$\frac{1}{4}$ .

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

Step 7.2

Multiply

$4$ by

$4$ .

$\frac{x e^{4 x}}{4} - \frac{1}{16} \int e^{u} d u$

Step 8

The integral of

$e^{u}$ with respect to

$u$ is

$e^{u}$ .

$\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u} + C)$

Step 9

Rewrite

$\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u} + C)$ as

$\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{u} + C$ .

$\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{u} + C$

Step 10

Replace all occurrences of

$u$ with

$4 x$ .

$\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{4 x} + C$