Calculus Examples

$\int x^{6} + e^{3 x} d x$

Step 1

Split the single integral into multiple integrals.

$\int x^{6} d x + \int e^{3 x} d x$

Step 2

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$\frac{1}{7} x^{7} + C + \int e^{3 x} d x$

Step 3

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 3.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 3.1.3

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

$3 \cdot 1$

Step 3.1.4

Multiply $3$ by $1$ .

$3$

Step 3.2

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

$\frac{1}{7} x^{7} + C + \int e^{u} \frac{1}{3} d u$

Step 4

Combine $e^{u}$ and $\frac{1}{3}$ .

$\frac{1}{7} x^{7} + C + \int \frac{e^{u}}{3} d u$

Step 5

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{7} x^{7} + C + \frac{1}{3} \int e^{u} d u$

Step 6

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

$\frac{1}{7} x^{7} + C + \frac{1}{3} (e^{u} + C)$

Step 7

Simplify.

$\frac{1}{7} x^{7} + \frac{1}{3} e^{u} + C$

Step 8

Replace all occurrences of $u$ with $3 x$ .

$\frac{1}{7} x^{7} + \frac{1}{3} e^{3 x} + C$