Calculus Examples

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

Step 1

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

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Step 1.1

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

$x$

$3$

$x^{3}$

$3 x^{2}$

$0 x$

$5$

Step 1.2

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$5$

Step 1.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$5$
			+	$x^{3}$	-	$3 x^{2}$

Step 1.4

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$5$
			-	$x^{3}$	+	$3 x^{2}$

Step 1.5

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$5$
			-	$x^{3}$	+	$3 x^{2}$
						$0$

Step 1.6

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$3 x^{2}$	+	$0 x$	+	$5$
			-	$x^{3}$	+	$3 x^{2}$
						$0$	+	$0 x$	+	$5$

Step 1.7

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

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3} + C + \int \frac{5}{x - 3} d x$

Step 4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$\frac{1}{3} x^{3} + C + 5 \int \frac{1}{x - 3} d x$

Step 5

Let $u = x - 3$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = x - 3$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $x - 3$ .

$\frac{d}{d x} [x - 3]$

Step 5.1.2

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 5.1.3

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

$1 + \frac{d}{d x} [- 3]$

Step 5.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$1 + 0$

Step 5.1.5

Add $1$ and $0$ .

$1$

Step 5.2

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

$\frac{1}{3} x^{3} + C + 5 \int \frac{1}{u} d u$

Step 6

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

$\frac{1}{3} x^{3} + C + 5 (\ln (| u |) + C)$

Step 7

Simplify.

$\frac{1}{3} x^{3} + 5 \ln (| u |) + C$

Step 8

Replace all occurrences of $u$ with $x - 3$ .

$\frac{1}{3} x^{3} + 5 \ln (| x - 3 |) + C$