Calculus Examples

$\int \frac{1}{a x + b} d x$

Step 1

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

Tap for more steps...

Step 1.1

Let $u = a x + b$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $a x + b$ .

$\frac{d}{d x} [a x + b]$

Step 1.1.2

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

$\frac{d}{d x} [a x] + \frac{d}{d x} [b]$

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

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

$a \frac{d}{d x} [x] + \frac{d}{d x} [b]$

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 $n = 1$ .

$a \cdot 1 + \frac{d}{d x} [b]$

Step 1.1.3.3

Multiply $a$ by $1$ .

$a + \frac{d}{d x} [b]$

Step 1.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.4.1

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

$a + 0$

Step 1.1.4.2

Add $a$ and $0$ .

$a$

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify.

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

Step 5.2

Combine $\frac{1}{a}$ and $\ln (| u |)$ .

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

Step 6

Replace all occurrences of $u$ with $a x + b$ .

$\frac{\ln (| a x + b |)}{a} + C$