Calculus Examples

$\int (7 x + e^{4 x}) d x$

Step 1

Remove parentheses.

$\int 7 x + e^{4 x} d x$

Step 2

Split the single integral into multiple integrals.

$\int 7 x d x + \int e^{4 x} d x$

Step 3

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int x d x + \int e^{4 x} d x$

Step 4

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$7 (\frac{1}{2} x^{2} + C) + \int e^{4 x} d x$

Step 5

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 5.1

Let $u = 4 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $4 x$ .

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

Step 5.1.2

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 5.1.3

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 5.1.4

Multiply $4$ by $1$ .

$4$

Step 5.2

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

$7 (\frac{1}{2} x^{2} + C) + \int e^{u} \frac{1}{4} d u$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$7 (\frac{x^{2}}{2} + C) + \int e^{u} \frac{1}{4} d u$

Step 6.2

Combine $e^{u}$ and $\frac{1}{4}$ .

$7 (\frac{x^{2}}{2} + C) + \int \frac{e^{u}}{4} d u$

Step 7

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$7 (\frac{x^{2}}{2} + C) + \frac{1}{4} \int e^{u} d u$

Step 8

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

$7 (\frac{x^{2}}{2} + C) + \frac{1}{4} (e^{u} + C)$

Step 9

Simplify.

$\frac{7 x^{2}}{2} + \frac{1}{4} e^{u} + C$

Step 10

Replace all occurrences of $u$ with $4 x$ .

$\frac{7 x^{2}}{2} + \frac{1}{4} e^{4 x} + C$

Step 11

Reorder terms.

$\frac{7}{2} x^{2} + \frac{1}{4} e^{4 x} + C$