Calculus Examples

$\int \frac{6 x^{3}}{2 x^{4} + 1} d x$

Step 1

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

$6 \int \frac{x^{3}}{2 x^{4} + 1} d x$

Step 2

Rewrite $x^{4}$ as ${(x^{4})}^{1}$ .

$6 \int \frac{x^{3}}{2 {(x^{4})}^{1} + 1} d x$

Step 3

Let $u_{1} = x^{4}$ . Then $d u_{1} = 4 x^{3} d x$ , so $\frac{1}{4} d u_{1} = x^{3} d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 3.1

Let $u_{1} = x^{4}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 3.1.1

Differentiate $x^{4}$ .

$\frac{d}{d x} [x^{4}]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3}$

Step 3.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$6 \int \frac{1}{2 {u_{1}}^{1} + 1} \cdot \frac{1}{4} d u_{1}$

Step 4

Simplify.

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Step 4.1

Simplify.

$6 \int \frac{1}{2 u_{1} + 1} \cdot \frac{1}{4} d u_{1}$

Step 4.2

Multiply $\frac{1}{2 u_{1} + 1}$ by $\frac{1}{4}$ .

$6 \int \frac{1}{(2 u_{1} + 1) \cdot 4} d u_{1}$

Step 4.3

Move $4$ to the left of $2 u_{1} + 1$ .

$6 \int \frac{1}{4 (2 u_{1} + 1)} d u_{1}$

Step 5

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

$6 (\frac{1}{4} \int \frac{1}{2 u_{1} + 1} d u_{1})$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{4}$ and $6$ .

$\frac{6}{4} \int \frac{1}{2 u_{1} + 1} d u_{1}$

Step 6.2

Cancel the common factor of $6$ and $4$ .

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Step 6.2.1

Factor $2$ out of $6$ .

$\frac{2 (3)}{4} \int \frac{1}{2 u_{1} + 1} d u_{1}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 3}{2 \cdot 2} \int \frac{1}{2 u_{1} + 1} d u_{1}$

Step 6.2.2.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 \cdot 2} \int \frac{1}{2 u_{1} + 1} d u_{1}$

Step 6.2.2.3

Rewrite the expression.

$\frac{3}{2} \int \frac{1}{2 u_{1} + 1} d u_{1}$

Step 7

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

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Step 7.1

Let $u_{2} = 2 u_{1} + 1$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 7.1.1

Differentiate $2 u_{1} + 1$ .

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

Step 7.1.2

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

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

Step 7.1.3

Evaluate $\frac{d}{d u_{1}} [2 u_{1}]$ .

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Step 7.1.3.1

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

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

Step 7.1.3.2

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

$2 \cdot 1 + \frac{d}{d u_{1}} [1]$

Step 7.1.3.3

Multiply $2$ by $1$ .

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

Step 7.1.4

Differentiate using the Constant Rule.

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Step 7.1.4.1

Since $1$ is constant with respect to $u_{1}$ , the derivative of $1$ with respect to $u_{1}$ is $0$ .

$2 + 0$

Step 7.1.4.2

Add $2$ and $0$ .

$2$

Step 7.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{3}{2} \int \frac{1}{u_{2}} \cdot \frac{1}{2} d u_{2}$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{u_{2}}$ by $\frac{1}{2}$ .

$\frac{3}{2} \int \frac{1}{u_{2} \cdot 2} d u_{2}$

Step 8.2

Move $2$ to the left of $u_{2}$ .

$\frac{3}{2} \int \frac{1}{2 u_{2}} d u_{2}$

Step 9

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

$\frac{3}{2} (\frac{1}{2} \int \frac{1}{u_{2}} d u_{2})$

Step 10

Simplify.

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Step 10.1

Multiply $\frac{1}{2}$ by $\frac{3}{2}$ .

$\frac{3}{2 \cdot 2} \int \frac{1}{u_{2}} d u_{2}$

Step 10.2

Multiply $2$ by $2$ .

$\frac{3}{4} \int \frac{1}{u_{2}} d u_{2}$

Step 11

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\frac{3}{4} (\ln (| u_{2} |) + C)$

Step 12

Simplify.

$\frac{3}{4} \ln (| u_{2} |) + C$

Step 13

Substitute back in for each integration substitution variable.

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Step 13.1

Replace all occurrences of $u_{2}$ with $2 u_{1} + 1$ .

$\frac{3}{4} \ln (| 2 u_{1} + 1 |) + C$

Step 13.2

Replace all occurrences of $u_{1}$ with $x^{4}$ .

$\frac{3}{4} \ln (| 2 x^{4} + 1 |) + C$