Calculus Examples

$\int x arccot (x) d x$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify the expression.

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Step 4.1

Simplify.

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Step 4.1.1

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 4.1.2

Move the negative in front of the fraction.

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

Step 4.1.3

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

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

Step 4.1.4

Multiply $- 1$ by $- 1$ .

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

Step 4.1.5

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

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

Step 4.2

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

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

Step 5

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

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Step 5.1

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 5.2

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 5.3

Multiply the new quotient term by the divisor.

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

Step 5.4

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 5.5

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 5.6

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

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

Apply the constant rule.

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

Step 8

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

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

Step 9

Simplify the expression.

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Step 9.1

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

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

Step 9.2

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

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

Step 10

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

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

Step 11

Simplify.

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

Step 12

Reorder terms.

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