Calculus Examples

$\int \frac{x - 2}{3 x + 6} d x$

Step 1

Divide $x - 2$ by $3 x + 6$ .

Tap for more steps...

Step 1.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$3 x$

$6$

$x$

$2$

Step 1.2

Divide the highest order term in the dividend $x$ by the highest order term in divisor $3 x$ .

					$\frac{1}{3}$
	$3 x$	+	$6$		$x$	-	$2$

Step 1.3

Multiply the new quotient term by the divisor.

				$\frac{1}{3}$
$3 x$	+	$6$		$x$	-	$2$
			+	$x$	+	$2$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $x + 2$

				$\frac{1}{3}$
$3 x$	+	$6$		$x$	-	$2$
			-	$x$	-	$2$

Step 1.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$\frac{1}{3}$
$3 x$	+	$6$		$x$	-	$2$
			-	$x$	-	$2$
					-	$4$

Step 1.6

The final answer is the quotient plus the remainder over the divisor.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

$\frac{1}{3} x + C + \int - \frac{4}{3 x + 6} d x$

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{3} x + C - \int \frac{4}{3 x + 6} d x$

Step 5

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\frac{1}{3} x + C - (4 \int \frac{1}{3 x + 6} d x)$

Step 6

Multiply $4$ by $- 1$ .

$\frac{1}{3} x + C - 4 \int \frac{1}{3 x + 6} d x$

Step 7

Let $u = 3 x + 6$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 7.1

Let $u = 3 x + 6$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 7.1.1

Differentiate $3 x + 6$ .

$\frac{d}{d x} [3 x + 6]$

Step 7.1.2

By the Sum Rule, the derivative of $3 x + 6$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [6]$

Step 7.1.3

Evaluate $\frac{d}{d x} [3 x]$ .

Tap for more steps...

Step 7.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 7.1.3.2

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

$3 \cdot 1 + \frac{d}{d x} [6]$

Step 7.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [6]$

Step 7.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 7.1.4.1

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$3 + 0$

Step 7.1.4.2

Add $3$ and $0$ .

$3$

Step 7.2

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

$\frac{1}{3} x + C - 4 \int \frac{1}{u} \cdot \frac{1}{3} d u$

Step 8

Simplify.

Tap for more steps...

Step 8.1

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

$\frac{1}{3} x + C - 4 \int \frac{1}{u \cdot 3} d u$

Step 8.2

Move $3$ to the left of $u$ .

$\frac{1}{3} x + C - 4 \int \frac{1}{3 u} d u$

Step 9

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} x + C - 4 (\frac{1}{3} \int \frac{1}{u} d u)$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Combine $\frac{1}{3}$ and $- 4$ .

$\frac{1}{3} x + C + \frac{- 4}{3} \int \frac{1}{u} d u$

Step 10.2

Move the negative in front of the fraction.

$\frac{1}{3} x + C - \frac{4}{3} \int \frac{1}{u} d u$

Step 11

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

$\frac{1}{3} x + C - \frac{4}{3} (\ln (| u |) + C)$

Step 12

Simplify.

$\frac{1}{3} x - \frac{4}{3} \ln (| u |) + C$

Step 13

Replace all occurrences of $u$ with $3 x + 6$ .

$\frac{1}{3} x - \frac{4}{3} \ln (| 3 x + 6 |) + C$