Calculus Examples

$\int \frac{4 x^{4} + 3}{4 x^{5} + 15 x + 2} d x$

Step 1

Let $u = 4 x^{5} + 15 x + 2$ . Then $d u = (20 x^{4} + 15) d x$ , so $\frac{1}{5} d u = (4 x^{4} + 3) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 x^{5} + 15 x + 2$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 x^{5} + 15 x + 2$ .

$\frac{d}{d x} [4 x^{5} + 15 x + 2]$

Step 1.1.2

By the Sum Rule, the derivative of $4 x^{5} + 15 x + 2$ with respect to $x$ is $\frac{d}{d x} [4 x^{5}] + \frac{d}{d x} [15 x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [4 x^{5}] + \frac{d}{d x} [15 x] + \frac{d}{d x} [2]$

Step 1.1.3

Evaluate $\frac{d}{d x} [4 x^{5}]$ .

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{5}$ with respect to $x$ is $4 \frac{d}{d x} [x^{5}]$ .

$4 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [15 x] + \frac{d}{d x} [2]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$4 (5 x^{4}) + \frac{d}{d x} [15 x] + \frac{d}{d x} [2]$

Step 1.1.3.3

Multiply $5$ by $4$ .

$20 x^{4} + \frac{d}{d x} [15 x] + \frac{d}{d x} [2]$

Step 1.1.4

Evaluate $\frac{d}{d x} [15 x]$ .

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Step 1.1.4.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x$ with respect to $x$ is $15 \frac{d}{d x} [x]$ .

$20 x^{4} + 15 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 1.1.4.2

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

$20 x^{4} + 15 \cdot 1 + \frac{d}{d x} [2]$

Step 1.1.4.3

Multiply $15$ by $1$ .

$20 x^{4} + 15 + \frac{d}{d x} [2]$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$20 x^{4} + 15 + 0$

Step 1.1.5.2

Add $20 x^{4} + 15$ and $0$ .

$20 x^{4} + 15$

Step 1.2

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

$\int \frac{1}{u} \cdot \frac{1}{5} d u$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{1}{u}$ by $\frac{1}{5}$ .

$\int \frac{1}{u \cdot 5} d u$

Step 2.2

Move $5$ to the left of $u$ .

$\int \frac{1}{5 u} d u$

Step 3

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{5} \int \frac{1}{u} d u$

Step 4

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

$\frac{1}{5} (\ln (| u |) + C)$

Step 5

Simplify.

$\frac{1}{5} \ln (| u |) + C$

Step 6

Replace all occurrences of $u$ with $4 x^{5} + 15 x + 2$ .

$\frac{1}{5} \ln (| 4 x^{5} + 15 x + 2 |) + C$