Calculus Examples

$\int \frac{11 x}{x - 10} d x$

Step 1

Since $11$ is constant with respect to $x$ , move $11$ out of the integral.

$11 \int \frac{x}{x - 10} d x$

Step 2

Divide $x$ by $x - 10$ .

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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 $0$ .

$x$

$10$

$x$

$0$

Step 2.2

Divide the highest order term in the dividend $x$ by the highest order term in divisor $x$ .

					$1$
	$x$	-	$10$		$x$	+	$0$

Step 2.3

Multiply the new quotient term by the divisor.

				$1$
$x$	-	$10$		$x$	+	$0$
			+	$x$	-	$10$

Step 2.4

The expression needs to be subtracted from the dividend, so change all the signs in $x - 10$

				$1$
$x$	-	$10$		$x$	+	$0$
			-	$x$	+	$10$

Step 2.5

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

				$1$
$x$	-	$10$		$x$	+	$0$
			-	$x$	+	$10$
					+	$10$

Step 2.6

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

$11 \int 1 + \frac{10}{x - 10} d x$

Step 3

Split the single integral into multiple integrals.

$11 (\int d x + \int \frac{10}{x - 10} d x)$

Step 4

Apply the constant rule.

$11 (x + C + \int \frac{10}{x - 10} d x)$

Step 5

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$11 (x + C + 10 \int \frac{1}{x - 10} d x)$

Step 6

Let $u = x - 10$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = x - 10$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $x - 10$ .

$\frac{d}{d x} [x - 10]$

Step 6.1.2

By the Sum Rule, the derivative of $x - 10$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 10]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 10]$

Step 6.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 10]$

Step 6.1.4

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10$ with respect to $x$ is $0$ .

$1 + 0$

Step 6.1.5

Add $1$ and $0$ .

$1$

Step 6.2

Rewrite the problem using $u$ and $d u$ .

$11 (x + C + 10 \int \frac{1}{u} d u)$

Step 7

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$11 (x + C + 10 (\ln (| u |) + C))$

Step 8

Simplify.

$11 (x + 10 \ln (| u |)) + C$

Step 9

Replace all occurrences of $u$ with $x - 10$ .

$11 (x + 10 \ln (| x - 10 |)) + C$