Calculus Examples

$\int_{0}^{10} \frac{24 t}{2 t + 3} d t$

Step 1

Since $24$ is constant with respect to $t$ , move $24$ out of the integral.

$24 \int_{0}^{10} \frac{t}{2 t + 3} d t$

Step 2

Divide $t$ by $2 t + 3$ .

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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 $0$ .

$2 t$

$3$

$t$

$0$

Step 2.2

Divide the highest order term in the dividend $t$ by the highest order term in divisor $2 t$ .

					$\frac{1}{2}$
	$2 t$	+	$3$		$t$	+	$0$

Step 2.3

Multiply the new quotient term by the divisor.

				$\frac{1}{2}$
$2 t$	+	$3$		$t$	+	$0$
			+	$t$	+	$\frac{3}{2}$

Step 2.4

The expression needs to be subtracted from the dividend, so change all the signs in $t + \frac{3}{2}$

				$\frac{1}{2}$
$2 t$	+	$3$		$t$	+	$0$
			-	$t$	-	$\frac{3}{2}$

Step 2.5

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

				$\frac{1}{2}$
$2 t$	+	$3$		$t$	+	$0$
			-	$t$	-	$\frac{3}{2}$
					-	$\frac{3}{2}$

Step 2.6

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

$24 \int_{0}^{10} \frac{1}{2} - \frac{3}{2 (2 t + 3)} d t$

Step 3

Split the single integral into multiple integrals.

$24 (\int_{0}^{10} \frac{1}{2} d t + \int_{0}^{10} - \frac{3}{2 (2 t + 3)} d t)$

Step 4

Apply the constant rule.

$24 (\frac{1}{2} t]_{0}^{10} + \int_{0}^{10} - \frac{3}{2 (2 t + 3)} d t)$

Step 5

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

$24 (\frac{t}{2}]_{0}^{10} + \int_{0}^{10} - \frac{3}{2 (2 t + 3)} d t)$

Step 6

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

$24 (\frac{t}{2}]_{0}^{10} - \int_{0}^{10} \frac{3}{2 (2 t + 3)} d t)$

Step 7

Since $\frac{3}{2}$ is constant with respect to $t$ , move $\frac{3}{2}$ out of the integral.

$24 (\frac{t}{2}]_{0}^{10} - (\frac{3}{2} \int_{0}^{10} \frac{1}{2 t + 3} d t))$

Step 8

Let $u = 2 t + 3$ . Then $d u = 2 d t$ , so $\frac{1}{2} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 8.1

Let $u = 2 t + 3$ . Find $\frac{d u}{d t}$ .

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Step 8.1.1

Differentiate $2 t + 3$ .

$\frac{d}{d t} [2 t + 3]$

Step 8.1.2

By the Sum Rule, the derivative of $2 t + 3$ with respect to $t$ is $\frac{d}{d t} [2 t] + \frac{d}{d t} [3]$ .

$\frac{d}{d t} [2 t] + \frac{d}{d t} [3]$

Step 8.1.3

Evaluate $\frac{d}{d t} [2 t]$ .

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Step 8.1.3.1

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t] + \frac{d}{d t} [3]$

Step 8.1.3.2

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

$2 \cdot 1 + \frac{d}{d t} [3]$

Step 8.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d t} [3]$

Step 8.1.4

Differentiate using the Constant Rule.

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Step 8.1.4.1

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

$2 + 0$

Step 8.1.4.2

Add $2$ and $0$ .

$2$

Step 8.2

Substitute the lower limit in for $t$ in $u = 2 t + 3$ .

$u_{lower} = 2 \cdot 0 + 3$

Step 8.3

Simplify.

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Step 8.3.1

Multiply $2$ by $0$ .

$u_{lower} = 0 + 3$

Step 8.3.2

Add $0$ and $3$ .

$u_{lower} = 3$

Step 8.4

Substitute the upper limit in for $t$ in $u = 2 t + 3$ .

$u_{upper} = 2 \cdot 10 + 3$

Step 8.5

Simplify.

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Step 8.5.1

Multiply $2$ by $10$ .

$u_{upper} = 20 + 3$

Step 8.5.2

Add $20$ and $3$ .

$u_{upper} = 23$

Step 8.6

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

$u_{lower} = 3$

$u_{upper} = 23$

Step 8.7

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

$24 (\frac{t}{2}]_{0}^{10} - \frac{3}{2} \int_{3}^{23} \frac{1}{u} \cdot \frac{1}{2} d u)$

Step 9

Simplify.

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Step 9.1

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

$24 (\frac{t}{2}]_{0}^{10} - \frac{3}{2} \int_{3}^{23} \frac{1}{u \cdot 2} d u)$

Step 9.2

Move $2$ to the left of $u$ .

$24 (\frac{t}{2}]_{0}^{10} - \frac{3}{2} \int_{3}^{23} \frac{1}{2 u} d u)$

Step 10

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

$24 (\frac{t}{2}]_{0}^{10} - \frac{3}{2} (\frac{1}{2} \int_{3}^{23} \frac{1}{u} d u))$

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{2}$ by $\frac{3}{2}$ .

$24 (\frac{t}{2}]_{0}^{10} - \frac{3}{2 \cdot 2} \int_{3}^{23} \frac{1}{u} d u)$

Step 11.2

Multiply $2$ by $2$ .

$24 (\frac{t}{2}]_{0}^{10} - \frac{3}{4} \int_{3}^{23} \frac{1}{u} d u)$

Step 12

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

$24 (\frac{t}{2}]_{0}^{10} - \frac{3}{4} \ln (| u |)]_{3}^{23})$

Step 13

Simplify.

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Step 13.1

Combine $\ln (| u |)]_{3}^{23}$ and $\frac{3}{4}$ .

$24 (\frac{t}{2}]_{0}^{10} - \frac{\ln (| u |)]_{3}^{23} \cdot 3}{4})$

Step 13.2

Move $3$ to the left of $\ln (| u |)]_{3}^{23}$ .

$24 (\frac{t}{2}]_{0}^{10} - \frac{3 (\ln (| u |)]_{3}^{23})}{4})$

Step 14

Substitute and simplify.

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Step 14.1

Evaluate $\frac{t}{2}$ at $10$ and at $0$ .

$24 ((\frac{10}{2}) - \frac{0}{2} - \frac{3 (\ln (| u |)]_{3}^{23})}{4})$

Step 14.2

Evaluate $\ln (| u |)$ at $23$ and at $3$ .

$24 ((\frac{10}{2}) - \frac{0}{2} - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3

Simplify.

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Step 14.3.1

Cancel the common factor of $10$ and $2$ .

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Step 14.3.1.1

Factor $2$ out of $10$ .

$24 (\frac{2 \cdot 5}{2} - \frac{0}{2} - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3.1.2

Cancel the common factors.

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Step 14.3.1.2.1

Factor $2$ out of $2$ .

$24 (\frac{2 \cdot 5}{2 (1)} - \frac{0}{2} - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3.1.2.2

Cancel the common factor.

$24 (\frac{2 \cdot 5}{2 \cdot 1} - \frac{0}{2} - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3.1.2.3

Rewrite the expression.

$24 (\frac{5}{1} - \frac{0}{2} - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3.1.2.4

Divide $5$ by $1$ .

$24 (5 - \frac{0}{2} - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3.2

Cancel the common factor of $0$ and $2$ .

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Step 14.3.2.1

Factor $2$ out of $0$ .

$24 (5 - \frac{2 (0)}{2} - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3.2.2

Cancel the common factors.

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Step 14.3.2.2.1

Factor $2$ out of $2$ .

$24 (5 - \frac{2 \cdot 0}{2 \cdot 1} - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3.2.2.2

Cancel the common factor.

$24 (5 - \frac{2 \cdot 0}{2 \cdot 1} - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3.2.2.3

Rewrite the expression.

$24 (5 - \frac{0}{1} - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3.2.2.4

Divide $0$ by $1$ .

$24 (5 - 0 - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3.3

Multiply $- 1$ by $0$ .

$24 (5 + 0 - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3.4

Add $5$ and $0$ .

$24 (5 - \frac{3 ((\ln (| 23 |)) - \ln (| 3 |))}{4})$

Step 14.3.5

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$24 (5 \cdot \frac{4}{4} - \frac{3 (\ln (| 23 |) - \ln (| 3 |))}{4})$

Step 14.3.6

Combine $5$ and $\frac{4}{4}$ .

$24 (\frac{5 \cdot 4}{4} - \frac{3 (\ln (| 23 |) - \ln (| 3 |))}{4})$

Step 14.3.7

Combine the numerators over the common denominator.

$24 \frac{5 \cdot 4 - 3 (\ln (| 23 |) - \ln (| 3 |))}{4}$

Step 14.3.8

Multiply $5$ by $4$ .

$24 \frac{20 - 3 (\ln (| 23 |) - \ln (| 3 |))}{4}$

Step 14.3.9

Combine $24$ and $\frac{20 - 3 (\ln (| 23 |) - \ln (| 3 |))}{4}$ .

$\frac{24 (20 - 3 (\ln (| 23 |) - \ln (| 3 |)))}{4}$

Step 14.3.10

Cancel the common factor of $24$ and $4$ .

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Step 14.3.10.1

Factor $4$ out of $24 (20 - 3 (\ln (| 23 |) - \ln (| 3 |)))$ .

$\frac{4 (6 (20 - 3 (\ln (| 23 |) - \ln (| 3 |))))}{4}$

Step 14.3.10.2

Cancel the common factors.

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Step 14.3.10.2.1

Factor $4$ out of $4$ .

$\frac{4 (6 (20 - 3 (\ln (| 23 |) - \ln (| 3 |))))}{4 (1)}$

Step 14.3.10.2.2

Cancel the common factor.

$\frac{4 (6 (20 - 3 (\ln (| 23 |) - \ln (| 3 |))))}{4 \cdot 1}$

Step 14.3.10.2.3

Rewrite the expression.

$\frac{6 (20 - 3 (\ln (| 23 |) - \ln (| 3 |)))}{1}$

Step 14.3.10.2.4

Divide $6 (20 - 3 (\ln (| 23 |) - \ln (| 3 |)))$ by $1$ .

$6 (20 - 3 (\ln (| 23 |) - \ln (| 3 |)))$

Step 15

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

$6 (20 - 3 \ln (\frac{| 23 |}{| 3 |}))$

Step 16

Simplify.

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Step 16.1

The absolute value is the distance between a number and zero. The distance between $0$ and $23$ is $23$ .

$6 (20 - 3 \ln (\frac{23}{| 3 |}))$

Step 16.2

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$6 (20 - 3 \ln (\frac{23}{3}))$

Step 16.3

Apply the distributive property.

$6 \cdot 20 + 6 (- 3 \ln (\frac{23}{3}))$

Step 16.4

Multiply $6$ by $20$ .

$120 + 6 (- 3 \ln (\frac{23}{3}))$

Step 16.5

Multiply $- 3$ by $6$ .

$120 - 18 \ln (\frac{23}{3})$

Step 17

The result can be shown in multiple forms.

Exact Form:

$120 - 18 \ln (\frac{23}{3})$

Decimal Form:

$83.33612530 \dots$

Step 18