Calculus Examples

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

Step 1

Move $2$ to the left of ${(x^{2} + 1)}^{2}$ .

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

Step 2

Let $u = x^{2} + 1$ . Then $d u = 2 x d x$ , so $\frac{1}{2 x} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.1

Let $u = x^{2} + 1$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.1.1

Differentiate $x^{2} + 1$ .

$\frac{d}{d x} [x^{2} + 1]$

Step 2.1.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [1]$

Step 2.1.4

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

$2 x + 0$

Step 2.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.2

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

$\int u^{2} d u$

Step 3

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{1}{3} u^{3} + C$

Step 4

Replace all occurrences of $u$ with $x^{2} + 1$ .

$\frac{1}{3} {(x^{2} + 1)}^{3} + C$