Calculus Examples

$\int (5 e^{2 x} + \frac{1}{x}) d x$

Step 1

Remove parentheses.

$\int 5 e^{2 x} + \frac{1}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int 5 e^{2 x} d x + \int \frac{1}{x} d x$

Step 3

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int e^{2 x} d x + \int \frac{1}{x} d x$

Step 4

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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Step 4.1

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

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Step 4.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 4.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 4.1.4

Multiply $2$ by $1$ .

$2$

Step 4.2

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

$5 \int e^{u} \frac{1}{2} d u + \int \frac{1}{x} d x$

Step 5

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

$5 \int \frac{e^{u}}{2} d u + \int \frac{1}{x} d x$

Step 6

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

$5 (\frac{1}{2} \int e^{u} d u) + \int \frac{1}{x} d x$

Step 7

Combine $\frac{1}{2}$ and $5$ .

$\frac{5}{2} \int e^{u} d u + \int \frac{1}{x} d x$

Step 8

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

$\frac{5}{2} (e^{u} + C) + \int \frac{1}{x} d x$

Step 9

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{5}{2} (e^{u} + C) + \ln (| x |) + C$

Step 10

Simplify.

$\frac{5}{2} e^{u} + \ln (| x |) + C$

Step 11

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

$\frac{5}{2} e^{2 x} + \ln (| x |) + C$