Calculus Examples

$\int x {(x + 1)}^{3} d x$

Step 1

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $x + 1$ .

$\frac{d}{d x} [x + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 1.1.3

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

$1 + \frac{d}{d x} [1]$

Step 1.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

$\int (u - 1) u^{3} d u$

Step 2

Multiply $(u - 1) u^{3}$ .

$\int u \cdot u^{3} - 1 u^{3} d u$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply $u$ by $u^{3}$ by adding the exponents.

Tap for more steps...

Step 3.1.1

Multiply $u$ by $u^{3}$ .

Tap for more steps...

Step 3.1.1.1

Raise $u$ to the power of $1$ .

$\int u^{1} u^{3} - 1 u^{3} d u$

Step 3.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int u^{1 + 3} - 1 u^{3} d u$

Step 3.1.2

Add $1$ and $3$ .

$\int u^{4} - 1 u^{3} d u$

Step 3.2

Rewrite $- 1 u^{3}$ as $- u^{3}$ .

$\int u^{4} - u^{3} d u$

Step 4

Split the single integral into multiple integrals.

$\int u^{4} d u + \int - u^{3} d u$

Step 5

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{5} u^{5} + C + \int - u^{3} d u$

Step 6

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$\frac{1}{5} u^{5} + C - \int u^{3} d u$

Step 7

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

$\frac{1}{5} u^{5} + C - (\frac{1}{4} u^{4} + C)$

Step 8

Simplify.

$\frac{1}{5} u^{5} - \frac{1}{4} u^{4} + C$

Step 9

Replace all occurrences of $u$ with $x + 1$ .

$\frac{1}{5} {(x + 1)}^{5} - \frac{1}{4} {(x + 1)}^{4} + C$