Calculus Examples

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

Step 1

Divide $x^{4} + x - 4$ by $x^{2} + 2$ .

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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 $0$ .

$x^{2}$

$0 x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$4$

Step 1.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$0 x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$4$

Step 1.3

Multiply the new quotient term by the divisor.

$x^{2}$

$0 x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$0$

$2 x^{2}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 0 + 2 x^{2}$

$x^{2}$

$0 x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$0$

$2 x^{2}$

Step 1.5

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

$x^{2}$

$0 x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$0$

$2 x^{2}$

Step 1.6

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

$x^{2}$

$0 x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$0$

$2 x^{2}$

$x$

$4$

Step 1.7

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

$x^{2}$

$0 x$

$2$

$x^{2}$

$0 x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$0$

$2 x^{2}$

$x$

$4$

Step 1.8

Multiply the new quotient term by the divisor.

						$x^{2}$	+	$0 x$	-	$2$
$x^{2}$	+	$0 x$	+	$2$		$x^{4}$	+	$0 x^{3}$	+	$0 x^{2}$	+	$x$	-	$4$
					-	$x^{4}$	-	$0$	-	$2 x^{2}$
									-	$2 x^{2}$	+	$x$	-	$4$
									-	$2 x^{2}$	+	$0$	-	$4$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} + 0 - 4$

						$x^{2}$	+	$0 x$	-	$2$
$x^{2}$	+	$0 x$	+	$2$		$x^{4}$	+	$0 x^{3}$	+	$0 x^{2}$	+	$x$	-	$4$
					-	$x^{4}$	-	$0$	-	$2 x^{2}$
									-	$2 x^{2}$	+	$x$	-	$4$
									+	$2 x^{2}$	-	$0$	+	$4$

Step 1.10

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

$x^{2}$

$0 x$

$2$

$x^{2}$

$0 x$

$2$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$4$

$x^{4}$

$0$

$2 x^{2}$

$x$

$4$

$2 x^{2}$

$0$

$4$

$x$

$0$

Step 1.11

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

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 4

Apply the constant rule.

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

Step 5

Let $u = x^{2} + 2$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

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

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Step 5.1.1

Differentiate $x^{2} + 2$ .

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

Step 5.1.2

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

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

Step 5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [2]$

Step 5.1.4

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

$2 x + 0$

Step 5.1.5

Add $2 x$ and $0$ .

$2 x$

Step 5.2

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

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

Step 6

Simplify.

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Step 6.1

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

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

Step 6.2

Move $2$ to the left of $u$ .

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

Step 7

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 8

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

$\frac{1}{3} x^{3} + C - 2 x + C + \frac{1}{2} (\ln (| u |) + C)$

Step 9

Simplify.

$\frac{1}{3} x^{3} - 2 x + \frac{1}{2} \ln (| u |) + C$

Step 10

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

$\frac{1}{3} x^{3} - 2 x + \frac{1}{2} \ln (| x^{2} + 2 |) + C$