Calculus Examples

$\int {(1 + x^{2})}^{2} \cdot 6 x d x$

Step 1

Move $6$ to the left of ${(1 + x^{2})}^{2}$ .

$\int 6 {(1 + x^{2})}^{2} x d x$

Step 2

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int {(1 + x^{2})}^{2} x d x$

Step 3

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

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Step 3.1

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

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Step 3.1.1

Differentiate $1 + x^{2}$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

$0 + \frac{d}{d x} [x^{2}]$

Step 3.1.4

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

$0 + 2 x$

Step 3.1.5

Add $0$ and $2 x$ .

$2 x$

Step 3.2

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

$6 \int u^{2} \frac{1}{2} d u$

Step 4

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

$6 \int \frac{u^{2}}{2} d u$

Step 5

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

$6 (\frac{1}{2} \int u^{2} d u)$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{2}$ and $6$ .

$\frac{6}{2} \int u^{2} d u$

Step 6.2

Cancel the common factor of $6$ and $2$ .

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Step 6.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 3}{2} \int u^{2} d u$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 3}{2 (1)} \int u^{2} d u$

Step 6.2.2.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 \cdot 1} \int u^{2} d u$

Step 6.2.2.3

Rewrite the expression.

$\frac{3}{1} \int u^{2} d u$

Step 6.2.2.4

Divide $3$ by $1$ .

$3 \int u^{2} d u$

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Rewrite $3 (\frac{1}{3} u^{3} + C)$ as $3 (\frac{1}{3}) u^{3} + C$ .

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

Step 8.2

Simplify.

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Step 8.2.1

Combine $3$ and $\frac{1}{3}$ .

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

Step 8.2.2

Cancel the common factor of $3$ .

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Step 8.2.2.1

Cancel the common factor.

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

Step 8.2.2.2

Rewrite the expression.

$1 u^{3} + C$

Step 8.2.3

Multiply $u^{3}$ by $1$ .

$u^{3} + C$

Step 9

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

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