Calculus Examples

$\int \frac{8 x}{3 + 4 x^{2}} d x$

Step 1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int \frac{x}{3 + 4 x^{2}} d x$

Step 2

Let $u = 3 + 4 x^{2}$ . Then $d u = 8 x d x$ , so $\frac{1}{8} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 3 + 4 x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $3 + 4 x^{2}$ .

$\frac{d}{d x} [3 + 4 x^{2}]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $3 + 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [4 x^{2}]$ .

$\frac{d}{d x} [3] + \frac{d}{d x} [4 x^{2}]$

Step 2.1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [4 x^{2}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 2.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$0 + 4 \frac{d}{d x} [x^{2}]$

Step 2.1.3.2

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

$0 + 4 (2 x)$

Step 2.1.3.3

Multiply $2$ by $4$ .

$0 + 8 x$

Step 2.1.4

Add $0$ and $8 x$ .

$8 x$

Step 2.2

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

$8 \int \frac{1}{u} \cdot \frac{1}{8} d u$

Step 3

Simplify.

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Step 3.1

Multiply $\frac{1}{u}$ by $\frac{1}{8}$ .

$8 \int \frac{1}{u \cdot 8} d u$

Step 3.2

Move $8$ to the left of $u$ .

$8 \int \frac{1}{8 u} d u$

Step 4

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$8 (\frac{1}{8} \int \frac{1}{u} d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{8}$ and $8$ .

$\frac{8}{8} \int \frac{1}{u} d u$

Step 5.2

Cancel the common factor of $8$ .

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Step 5.2.1

Cancel the common factor.

$\frac{8}{8} \int \frac{1}{u} d u$

Step 5.2.2

Rewrite the expression.

$1 \int \frac{1}{u} d u$

Step 5.3

Multiply $\int \frac{1}{u} d u$ by $1$ .

$\int \frac{1}{u} d u$

Step 6

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

$\ln (| u |) + C$

Step 7

Replace all occurrences of $u$ with $3 + 4 x^{2}$ .

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