Calculus Examples

$\int \frac{2 x^{3} - 5 x^{2} - 2 x - 3}{2 x^{2} + x + 1} d x$

Step 1

Divide $2 x^{3} - 5 x^{2} - 2 x - 3$ by $2 x^{2} + x + 1$ .

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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 $0$ .

$2 x^{2}$

$x$

$1$

$2 x^{3}$

$5 x^{2}$

$2 x$

$3$

Step 1.2

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $2 x^{2}$ .

$x$

$2 x^{2}$

$x$

$1$

$2 x^{3}$

$5 x^{2}$

$2 x$

$3$

Step 1.3

Multiply the new quotient term by the divisor.

$x$

$2 x^{2}$

$x$

$1$

$2 x^{3}$

$5 x^{2}$

$2 x$

$3$

$2 x^{3}$

$x^{2}$

$x$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} + x^{2} + x$

						$x$
$2 x^{2}$	+	$x$	+	$1$		$2 x^{3}$	-	$5 x^{2}$	-	$2 x$	-	$3$
					-	$2 x^{3}$	-	$x^{2}$	-	$x$

Step 1.5

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

						$x$
$2 x^{2}$	+	$x$	+	$1$		$2 x^{3}$	-	$5 x^{2}$	-	$2 x$	-	$3$
					-	$2 x^{3}$	-	$x^{2}$	-	$x$
							-	$6 x^{2}$	-	$3 x$

Step 1.6

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

						$x$
$2 x^{2}$	+	$x$	+	$1$		$2 x^{3}$	-	$5 x^{2}$	-	$2 x$	-	$3$
					-	$2 x^{3}$	-	$x^{2}$	-	$x$
							-	$6 x^{2}$	-	$3 x$	-	$3$

Step 1.7

Divide the highest order term in the dividend $- 6 x^{2}$ by the highest order term in divisor $2 x^{2}$ .

						$x$	-	$3$
$2 x^{2}$	+	$x$	+	$1$		$2 x^{3}$	-	$5 x^{2}$	-	$2 x$	-	$3$
					-	$2 x^{3}$	-	$x^{2}$	-	$x$
							-	$6 x^{2}$	-	$3 x$	-	$3$

Step 1.8

Multiply the new quotient term by the divisor.

						$x$	-	$3$
$2 x^{2}$	+	$x$	+	$1$		$2 x^{3}$	-	$5 x^{2}$	-	$2 x$	-	$3$
					-	$2 x^{3}$	-	$x^{2}$	-	$x$
							-	$6 x^{2}$	-	$3 x$	-	$3$
							-	$6 x^{2}$	-	$3 x$	-	$3$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} - 3 x - 3$

						$x$	-	$3$
$2 x^{2}$	+	$x$	+	$1$		$2 x^{3}$	-	$5 x^{2}$	-	$2 x$	-	$3$
					-	$2 x^{3}$	-	$x^{2}$	-	$x$
							-	$6 x^{2}$	-	$3 x$	-	$3$
							+	$6 x^{2}$	+	$3 x$	+	$3$

Step 1.10

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

						$x$	-	$3$
$2 x^{2}$	+	$x$	+	$1$		$2 x^{3}$	-	$5 x^{2}$	-	$2 x$	-	$3$
					-	$2 x^{3}$	-	$x^{2}$	-	$x$
							-	$6 x^{2}$	-	$3 x$	-	$3$
							+	$6 x^{2}$	+	$3 x$	+	$3$
												$0$

Step 1.11

Since the remander is $0$ , the final answer is the quotient.

$\int x - 3 d x$

Step 2

Split the single integral into multiple integrals.

$\int x d x + \int - 3 d x$

Step 3

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

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

Step 4

Apply the constant rule.

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

Step 5

Simplify.

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