Calculus Examples

$\frac{3 x^{2} + 5 x - 1}{x - 2}$

Step 1

Write $\frac{3 x^{2} + 5 x - 1}{x - 2}$ as a function.

$f (x) = \frac{3 x^{2} + 5 x - 1}{x - 2}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{3 x^{2} + 5 x - 1}{x - 2} d x$

Step 4

Divide $3 x^{2} + 5 x - 1$ by $x - 2$ .

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Step 4.1

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

$x$

$2$

$3 x^{2}$

$5 x$

$1$

Step 4.2

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $x$ .

					$3 x$
	$x$	-	$2$		$3 x^{2}$	+	$5 x$	-	$1$

Step 4.3

Multiply the new quotient term by the divisor.

				$3 x$
$x$	-	$2$		$3 x^{2}$	+	$5 x$	-	$1$
			+	$3 x^{2}$	-	$6 x$

Step 4.4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} - 6 x$

				$3 x$
$x$	-	$2$		$3 x^{2}$	+	$5 x$	-	$1$
			-	$3 x^{2}$	+	$6 x$

Step 4.5

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

				$3 x$
$x$	-	$2$		$3 x^{2}$	+	$5 x$	-	$1$
			-	$3 x^{2}$	+	$6 x$
					+	$11 x$

Step 4.6

Pull the next terms from the original dividend down into the current dividend.

				$3 x$
$x$	-	$2$		$3 x^{2}$	+	$5 x$	-	$1$
			-	$3 x^{2}$	+	$6 x$
					+	$11 x$	-	$1$

Step 4.7

Divide the highest order term in the dividend $11 x$ by the highest order term in divisor $x$ .

				$3 x$	+	$11$
$x$	-	$2$		$3 x^{2}$	+	$5 x$	-	$1$
			-	$3 x^{2}$	+	$6 x$
					+	$11 x$	-	$1$

Step 4.8

Multiply the new quotient term by the divisor.

				$3 x$	+	$11$
$x$	-	$2$		$3 x^{2}$	+	$5 x$	-	$1$
			-	$3 x^{2}$	+	$6 x$
					+	$11 x$	-	$1$
					+	$11 x$	-	$22$

Step 4.9

The expression needs to be subtracted from the dividend, so change all the signs in $11 x - 22$

				$3 x$	+	$11$
$x$	-	$2$		$3 x^{2}$	+	$5 x$	-	$1$
			-	$3 x^{2}$	+	$6 x$
					+	$11 x$	-	$1$
					-	$11 x$	+	$22$

Step 4.10

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

				$3 x$	+	$11$
$x$	-	$2$		$3 x^{2}$	+	$5 x$	-	$1$
			-	$3 x^{2}$	+	$6 x$
					+	$11 x$	-	$1$
					-	$11 x$	+	$22$
							+	$21$

Step 4.11

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

$\int 3 x + 11 + \frac{21}{x - 2} d x$

Step 5

Split the single integral into multiple integrals.

$\int 3 x d x + \int 11 d x + \int \frac{21}{x - 2} d x$

Step 6

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x d x + \int 11 d x + \int \frac{21}{x - 2} d x$

Step 7

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$3 (\frac{1}{2} x^{2} + C) + \int 11 d x + \int \frac{21}{x - 2} d x$

Step 8

Apply the constant rule.

$3 (\frac{1}{2} x^{2} + C) + 11 x + C + \int \frac{21}{x - 2} d x$

Step 9

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

$3 (\frac{x^{2}}{2} + C) + 11 x + C + \int \frac{21}{x - 2} d x$

Step 10

Since $21$ is constant with respect to $x$ , move $21$ out of the integral.

$3 (\frac{x^{2}}{2} + C) + 11 x + C + 21 \int \frac{1}{x - 2} d x$

Step 11

Let $u = x - 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 11.1

Let $u = x - 2$ . Find $\frac{d u}{d x}$ .

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Step 11.1.1

Differentiate $x - 2$ .

$\frac{d}{d x} [x - 2]$

Step 11.1.2

By the Sum Rule, the derivative of $x - 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 11.1.3

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

$1 + \frac{d}{d x} [- 2]$

Step 11.1.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$1 + 0$

Step 11.1.5

Add $1$ and $0$ .

$1$

Step 11.2

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

$3 (\frac{x^{2}}{2} + C) + 11 x + C + 21 \int \frac{1}{u} d u$

Step 12

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

$3 (\frac{x^{2}}{2} + C) + 11 x + C + 21 (\ln (| u |) + C)$

Step 13

Simplify.

$\frac{3 x^{2}}{2} + 11 x + 21 \ln (| u |) + C$

Step 14

Replace all occurrences of $u$ with $x - 2$ .

$\frac{3 x^{2}}{2} + 11 x + 21 \ln (| x - 2 |) + C$

Step 15

Reorder terms.

$\frac{3}{2} x^{2} + 11 x + 21 \ln (| x - 2 |) + C$

Step 16

The answer is the antiderivative of the function $f (x) = \frac{3 x^{2} + 5 x - 1}{x - 2}$ .

$F (x) =$ $\frac{3}{2} x^{2} + 11 x + 21 \ln (| x - 2 |) + C$