Calculus Examples

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

Step 1

Reorder $1$ and $x^{2}$ .

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

Step 2

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

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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 $0$ .

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$0$

Step 2.2

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

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

Step 2.3

Multiply the new quotient term by the divisor.

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

Step 2.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 0 + 1$

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

Step 2.5

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

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

Step 2.6

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Apply the constant rule.

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

Step 5

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

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

Step 6

Simplify the expression.

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Step 6.1

Reorder $x^{2}$ and $1$ .

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

Step 6.2

Rewrite $1$ as $1^{2}$ .

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

Step 7

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

$x + C - (\arctan (x) + C)$

Step 8

Simplify.

$x - \arctan (x) + C$