Calculus Examples

$\frac{x^{2}}{x^{2} + 5}$

Step 1

Divide $x^{2}$ by $x^{2} + 5$ .

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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 $0$ .

$x^{2}$

$0 x$

$5$

$x^{2}$

$0 x$

$0$

Step 1.2

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

							$1$
	$x^{2}$	+	$0 x$	+	$5$		$x^{2}$	+	$0 x$	+	$0$

Step 1.3

Multiply the new quotient term by the divisor.

						$1$
$x^{2}$	+	$0 x$	+	$5$		$x^{2}$	+	$0 x$	+	$0$
					+	$x^{2}$	+	$0$	+	$5$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 0 + 5$

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

Step 1.5

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

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

Step 1.6

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

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

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

Step 4

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

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

Step 5

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$x + C - (5 \int \frac{1}{x^{2} + 5} d x)$

Step 6

Simplify the expression.

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Step 6.1

Multiply $5$ by $- 1$ .

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

Step 6.2

Reorder $x^{2}$ and $5$ .

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

Step 7

Rewrite $5$ as ${\sqrt{5}}^{2}$ .

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

Step 8

The integral of $\frac{1}{{\sqrt{5}}^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{\sqrt{5}} \arctan (\frac{x}{\sqrt{5}}) + C$ .

$x + C - 5 (\frac{1}{\sqrt{5}} \arctan (\frac{x}{\sqrt{5}}) + C)$

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{\sqrt{5}}$ and $\arctan (\frac{x}{\sqrt{5}})$ .

$x + C - 5 (\frac{\arctan (\frac{x}{\sqrt{5}})}{\sqrt{5}} + C)$

Step 9.2

Simplify.

$x - 5 \frac{\arctan (\frac{x \sqrt{5}}{5})}{\sqrt{5}} + C$

Step 9.3

Simplify.

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Step 9.3.1

Combine $- 5$ and $\frac{\arctan (\frac{x \sqrt{5}}{5})}{\sqrt{5}}$ .

$x + \frac{- 5 \arctan (\frac{x \sqrt{5}}{5})}{\sqrt{5}} + C$

Step 9.3.2

Move the negative in front of the fraction.

$x - \frac{5 \arctan (\frac{x \sqrt{5}}{5})}{\sqrt{5}} + C$

Step 10

Reorder terms.

$x - \frac{5}{\sqrt{5}} \arctan (\frac{\sqrt{5}}{5} x) + C$