Calculus Examples

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

Step 1

Reorder $4$ and $x^{2}$ .

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

Step 2

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

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Step 2.1

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

$x^{2}$

$0 x$

$4$

$x^{2}$

$0 x$

$0$

Step 2.2

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

							$1$
	$x^{2}$	+	$0 x$	+	$4$		$x^{2}$	+	$0 x$	+	$0$

Step 2.3

Multiply the new quotient term by the divisor.

						$1$
$x^{2}$	+	$0 x$	+	$4$		$x^{2}$	+	$0 x$	+	$0$
					+	$x^{2}$	+	$0$	+	$4$

Step 2.4

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

						$1$
$x^{2}$	+	$0 x$	+	$4$		$x^{2}$	+	$0 x$	+	$0$
					-	$x^{2}$	-	$0$	-	$4$

Step 2.5

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

						$1$
$x^{2}$	+	$0 x$	+	$4$		$x^{2}$	+	$0 x$	+	$0$
					-	$x^{2}$	-	$0$	-	$4$
									-	$4$

Step 2.6

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Apply the constant rule.

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify the expression.

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Step 7.1

Multiply $4$ by $- 1$ .

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

Step 7.2

Reorder $x^{2}$ and $4$ .

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

Step 7.3

Rewrite $4$ as $2^{2}$ .

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

Step 8

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

$x + C - 4 (\frac{1}{2} \arctan (\frac{x}{2}) + C)$

Step 9

Simplify.

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Step 9.1

Combine $\frac{1}{2}$ and $\arctan (\frac{x}{2})$ .

$x + C - 4 (\frac{\arctan (\frac{x}{2})}{2} + C)$

Step 9.2

Simplify.

$x - 2 \arctan (\frac{x}{2}) + C$

Step 9.3

Reorder terms.

$x - 2 \arctan (\frac{1}{2} x) + C$