Calculus Examples

$\int (x^{3} - 3 x + e^{3 x}) d x$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{1}{4} x^{4} + C + \int - 3 x d x + \int e^{3 x} d x$

Step 4

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$\frac{1}{4} x^{4} + C - 3 \int x d x + \int e^{3 x} d x$

Step 5

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

$\frac{1}{4} x^{4} + C - 3 (\frac{1}{2} x^{2} + C) + \int e^{3 x} d x$

Step 6

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 6.1

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

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Step 6.1.1

Differentiate $3 x$ .

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

Step 6.1.2

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 6.1.3

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 6.1.4

Multiply $3$ by $1$ .

$3$

Step 6.2

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

$\frac{1}{4} x^{4} + C - 3 (\frac{1}{2} x^{2} + C) + \int e^{u} \frac{1}{3} d u$

Step 7

Simplify.

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Step 7.1

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

$\frac{1}{4} x^{4} + C - 3 (\frac{x^{2}}{2} + C) + \int e^{u} \frac{1}{3} d u$

Step 7.2

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

$\frac{1}{4} x^{4} + C - 3 (\frac{x^{2}}{2} + C) + \int \frac{e^{u}}{3} d u$

Step 8

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

$\frac{1}{4} x^{4} + C - 3 (\frac{x^{2}}{2} + C) + \frac{1}{3} \int e^{u} d u$

Step 9

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

$\frac{1}{4} x^{4} + C - 3 (\frac{x^{2}}{2} + C) + \frac{1}{3} (e^{u} + C)$

Step 10

Simplify.

$\frac{x^{4}}{4} - \frac{3 x^{2}}{2} + \frac{1}{3} e^{u} + C$

Step 11

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

$\frac{x^{4}}{4} - \frac{3 x^{2}}{2} + \frac{1}{3} e^{3 x} + C$

Step 12

Reorder terms.

$\frac{1}{4} x^{4} - \frac{3}{2} x^{2} + \frac{1}{3} e^{3 x} + C$