Calculus Examples

$\int x \cdot \arctan (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \arctan (x)$ and $d v = x$ .

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

Step 2

Simplify.

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Combine $\frac{1}{2}$ and $x^{2}$ .

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

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

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

Step 3

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

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

Step 4

Combine $x^{2}$ and $\frac{1}{x^{2} + 1}$ .

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

Step 5

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

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

$x^{2}$

$0 x$

$1$

$x^{2}$

$0 x$

$0$

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

Apply the constant rule.

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

Step 8

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

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

Step 9

Simplify the expression.

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

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

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

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

Step 10

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

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

Step 11

Simplify.

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

Step 12

Reorder terms.

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