Calculus Examples

$\int_{3}^{5} \frac{2 x^{2} + x + 4}{x - 1} d x$

Step 1

Divide $2 x^{2} + x + 4$ by $x - 1$ .

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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 $0$ .

$x$

$1$

$2 x^{2}$

$x$

$4$

Step 1.2

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

					$2 x$
	$x$	-	$1$		$2 x^{2}$	+	$x$	+	$4$

Step 1.3

Multiply the new quotient term by the divisor.

				$2 x$
$x$	-	$1$		$2 x^{2}$	+	$x$	+	$4$
			+	$2 x^{2}$	-	$2 x$

Step 1.4

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

				$2 x$
$x$	-	$1$		$2 x^{2}$	+	$x$	+	$4$
			-	$2 x^{2}$	+	$2 x$

Step 1.5

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

				$2 x$
$x$	-	$1$		$2 x^{2}$	+	$x$	+	$4$
			-	$2 x^{2}$	+	$2 x$
					+	$3 x$

Step 1.6

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

				$2 x$
$x$	-	$1$		$2 x^{2}$	+	$x$	+	$4$
			-	$2 x^{2}$	+	$2 x$
					+	$3 x$	+	$4$

Step 1.7

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

				$2 x$	+	$3$
$x$	-	$1$		$2 x^{2}$	+	$x$	+	$4$
			-	$2 x^{2}$	+	$2 x$
					+	$3 x$	+	$4$

Step 1.8

Multiply the new quotient term by the divisor.

				$2 x$	+	$3$
$x$	-	$1$		$2 x^{2}$	+	$x$	+	$4$
			-	$2 x^{2}$	+	$2 x$
					+	$3 x$	+	$4$
					+	$3 x$	-	$3$

Step 1.9

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

				$2 x$	+	$3$
$x$	-	$1$		$2 x^{2}$	+	$x$	+	$4$
			-	$2 x^{2}$	+	$2 x$
					+	$3 x$	+	$4$
					-	$3 x$	+	$3$

Step 1.10

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

				$2 x$	+	$3$
$x$	-	$1$		$2 x^{2}$	+	$x$	+	$4$
			-	$2 x^{2}$	+	$2 x$
					+	$3 x$	+	$4$
					-	$3 x$	+	$3$
							+	$7$

Step 1.11

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

$\int_{3}^{5} 2 x + 3 + \frac{7}{x - 1} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{3}^{5} 2 x d x + \int_{3}^{5} 3 d x + \int_{3}^{5} \frac{7}{x - 1} d x$

Step 3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{3}^{5} x d x + \int_{3}^{5} 3 d x + \int_{3}^{5} \frac{7}{x - 1} d x$

Step 4

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

$2 (\frac{1}{2} x^{2}]_{3}^{5}) + \int_{3}^{5} 3 d x + \int_{3}^{5} \frac{7}{x - 1} d x$

Step 5

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

$2 (\frac{x^{2}}{2}]_{3}^{5}) + \int_{3}^{5} 3 d x + \int_{3}^{5} \frac{7}{x - 1} d x$

Step 6

Apply the constant rule.

$2 (\frac{x^{2}}{2}]_{3}^{5}) + 3 x]_{3}^{5} + \int_{3}^{5} \frac{7}{x - 1} d x$

Step 7

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$2 (\frac{x^{2}}{2}]_{3}^{5}) + 3 x]_{3}^{5} + 7 \int_{3}^{5} \frac{1}{x - 1} d x$

Step 8

Let $u = x - 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = x - 1$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $x - 1$ .

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

Step 8.1.2

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 8.1.3

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

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

Step 8.1.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$1 + 0$

Step 8.1.5

Add $1$ and $0$ .

$1$

Step 8.2

Substitute the lower limit in for $x$ in $u = x - 1$ .

$u_{lower} = 3 - 1$

Step 8.3

Subtract $1$ from $3$ .

$u_{lower} = 2$

Step 8.4

Substitute the upper limit in for $x$ in $u = x - 1$ .

$u_{upper} = 5 - 1$

Step 8.5

Subtract $1$ from $5$ .

$u_{upper} = 4$

Step 8.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 2$

$u_{upper} = 4$

Step 8.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$2 (\frac{x^{2}}{2}]_{3}^{5}) + 3 x]_{3}^{5} + 7 \int_{2}^{4} \frac{1}{u} d u$

Step 9

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

$2 (\frac{x^{2}}{2}]_{3}^{5}) + 3 x]_{3}^{5} + 7 (\ln (| u |)]_{2}^{4})$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $3$ .

$2 ((\frac{5^{2}}{2}) - \frac{3^{2}}{2}) + 3 x]_{3}^{5} + 7 (\ln (| u |)]_{2}^{4})$

Step 10.2

Evaluate $3 x$ at $5$ and at $3$ .

$2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + 3 \cdot 5 - 3 \cdot 3 + 7 (\ln (| u |)]_{2}^{4})$

Step 10.3

Evaluate $\ln (| u |)$ at $4$ and at $2$ .

$2 (\frac{5^{2}}{2} - \frac{3^{2}}{2}) + 3 \cdot 5 - 3 \cdot 3 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4

Simplify.

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Step 10.4.1

Raise $5$ to the power of $2$ .

$2 (\frac{25}{2} - \frac{3^{2}}{2}) + 3 \cdot 5 - 3 \cdot 3 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.2

Raise $3$ to the power of $2$ .

$2 (\frac{25}{2} - \frac{9}{2}) + 3 \cdot 5 - 3 \cdot 3 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.3

Combine the numerators over the common denominator.

$2 \frac{25 - 9}{2} + 3 \cdot 5 - 3 \cdot 3 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.4

Subtract $9$ from $25$ .

$2 (\frac{16}{2}) + 3 \cdot 5 - 3 \cdot 3 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.5

Cancel the common factor of $16$ and $2$ .

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Step 10.4.5.1

Factor $2$ out of $16$ .

$2 \frac{2 \cdot 8}{2} + 3 \cdot 5 - 3 \cdot 3 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.5.2

Cancel the common factors.

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Step 10.4.5.2.1

Factor $2$ out of $2$ .

$2 \frac{2 \cdot 8}{2 (1)} + 3 \cdot 5 - 3 \cdot 3 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.5.2.2

Cancel the common factor.

$2 \frac{2 \cdot 8}{2 \cdot 1} + 3 \cdot 5 - 3 \cdot 3 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.5.2.3

Rewrite the expression.

$2 (\frac{8}{1}) + 3 \cdot 5 - 3 \cdot 3 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.5.2.4

Divide $8$ by $1$ .

$2 \cdot 8 + 3 \cdot 5 - 3 \cdot 3 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.6

Multiply $2$ by $8$ .

$16 + 3 \cdot 5 - 3 \cdot 3 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.7

Multiply $3$ by $5$ .

$16 + 15 - 3 \cdot 3 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.8

Multiply $- 3$ by $3$ .

$16 + 15 - 9 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.9

Subtract $9$ from $15$ .

$16 + 6 + 7 ((\ln (| 4 |)) - \ln (| 2 |))$

Step 10.4.10

Add $16$ and $6$ .

$22 + 7 (\ln (| 4 |) - \ln (| 2 |))$

Step 11

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$22 + 7 \ln (\frac{| 4 |}{| 2 |})$

Step 12

Simplify.

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Step 12.1

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$22 + 7 \ln (\frac{4}{| 2 |})$

Step 12.2

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$22 + 7 \ln (\frac{4}{2})$

Step 12.3

Divide $4$ by $2$ .

$22 + 7 \ln (2)$

Step 13

The result can be shown in multiple forms.

Exact Form:

$22 + 7 \ln (2)$

Decimal Form:

$26.85203026 \dots$

Step 14