Calculus Examples

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

Step 1

Divide $x^{2} - 3 x + 2$ by $x + 1$ .

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 $0$ .

$x$

$1$

$x^{2}$

$3 x$

$2$

Step 1.2

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x$ .

					$x$
	$x$	+	$1$		$x^{2}$	-	$3 x$	+	$2$

Step 1.3

Multiply the new quotient term by the divisor.

				$x$
$x$	+	$1$		$x^{2}$	-	$3 x$	+	$2$
			+	$x^{2}$	+	$x$

Step 1.4

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

				$x$
$x$	+	$1$		$x^{2}$	-	$3 x$	+	$2$
			-	$x^{2}$	-	$x$

Step 1.5

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

				$x$
$x$	+	$1$		$x^{2}$	-	$3 x$	+	$2$
			-	$x^{2}$	-	$x$
					-	$4 x$

Step 1.6

Pull the next terms from the original dividend down into the current dividend.

				$x$
$x$	+	$1$		$x^{2}$	-	$3 x$	+	$2$
			-	$x^{2}$	-	$x$
					-	$4 x$	+	$2$

Step 1.7

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $x$ .

				$x$	-	$4$
$x$	+	$1$		$x^{2}$	-	$3 x$	+	$2$
			-	$x^{2}$	-	$x$
					-	$4 x$	+	$2$

Step 1.8

Multiply the new quotient term by the divisor.

				$x$	-	$4$
$x$	+	$1$		$x^{2}$	-	$3 x$	+	$2$
			-	$x^{2}$	-	$x$
					-	$4 x$	+	$2$
					-	$4 x$	-	$4$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x - 4$

				$x$	-	$4$
$x$	+	$1$		$x^{2}$	-	$3 x$	+	$2$
			-	$x^{2}$	-	$x$
					-	$4 x$	+	$2$
					+	$4 x$	+	$4$

Step 1.10

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

				$x$	-	$4$
$x$	+	$1$		$x^{2}$	-	$3 x$	+	$2$
			-	$x^{2}$	-	$x$
					-	$4 x$	+	$2$
					+	$4 x$	+	$4$
							+	$6$

Step 1.11

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

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

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

Step 4

Apply the constant rule.

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

Step 5

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

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

Step 6

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 6.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 6.1.1

Differentiate $x + 1$ .

$\frac{d}{d x} [x + 1]$

Step 6.1.2

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

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 6.1.3

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

$1 + \frac{d}{d x} [1]$

Step 6.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0$

Step 6.1.5

Add $1$ and $0$ .

$1$

Step 6.2

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

$\frac{1}{2} x^{2} + C - 4 x + C + 6 \int \frac{1}{u} d u$

Step 7

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

$\frac{1}{2} x^{2} + C - 4 x + C + 6 (\ln (| u |) + C)$

Step 8

Simplify.

$\frac{1}{2} x^{2} - 4 x + 6 \ln (| u |) + C$

Step 9

Replace all occurrences of $u$ with $x + 1$ .

$\frac{1}{2} x^{2} - 4 x + 6 \ln (| x + 1 |) + C$