Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 4

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

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

Step 5

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Multiply $5$ by $- 1$ .

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

Step 9

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

$4 (\frac{2 x^{\frac{3}{2}}}{3}]_{1}^{2}) - 5 (\ln (| x |)]_{1}^{2})$

Step 10

Simplify the answer.

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Step 10.1

Substitute and simplify.

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Step 10.1.1

Evaluate $\frac{2 x^{\frac{3}{2}}}{3}$ at $2$ and at $1$ .

$4 ((\frac{2 \cdot 2^{\frac{3}{2}}}{3}) - \frac{2 \cdot 1^{\frac{3}{2}}}{3}) - 5 (\ln (| x |)]_{1}^{2})$

Step 10.1.2

Evaluate $\ln (| x |)$ at $2$ and at $1$ .

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

Step 10.1.3

Simplify.

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Step 10.1.3.1

Multiply $2$ by $2^{\frac{3}{2}}$ by adding the exponents.

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Step 10.1.3.1.1

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

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Step 10.1.3.1.1.1

Raise $2$ to the power of $1$ .

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

Step 10.1.3.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 10.1.3.1.2

Write $1$ as a fraction with a common denominator.

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

Step 10.1.3.1.3

Combine the numerators over the common denominator.

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

Step 10.1.3.1.4

Add $2$ and $3$ .

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

Step 10.1.3.2

One to any power is one.

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

Step 10.1.3.3

Multiply $2$ by $1$ .

$4 (\frac{2^{\frac{5}{2}}}{3} - \frac{2}{3}) - 5 (\ln (| 2 |) - \ln (| 1 |))$

Step 10.1.3.4

Combine the numerators over the common denominator.

$4 \frac{2^{\frac{5}{2}} - 2}{3} - 5 (\ln (| 2 |) - \ln (| 1 |))$

Step 10.1.3.5

Combine $4$ and $\frac{2^{\frac{5}{2}} - 2}{3}$ .

$\frac{4 (2^{\frac{5}{2}} - 2)}{3} - 5 (\ln (| 2 |) - \ln (| 1 |))$

Step 10.1.3.6

To write $- 5 (\ln (| 2 |) - \ln (| 1 |))$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 10.1.3.7

Combine $- 5 (\ln (| 2 |) - \ln (| 1 |))$ and $\frac{3}{3}$ .

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

Step 10.1.3.8

Combine the numerators over the common denominator.

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

Step 10.1.3.9

Multiply $3$ by $- 5$ .

$\frac{4 (2^{\frac{5}{2}} - 2) - 15 (\ln (| 2 |) - \ln (| 1 |))}{3}$

Step 10.2

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

$\frac{4 (2^{\frac{5}{2}} - 2) - 15 \ln (\frac{| 2 |}{| 1 |})}{3}$

Step 10.3

Simplify.

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Step 10.3.1

Apply the distributive property.

$\frac{4 \cdot 2^{\frac{5}{2}} + 4 \cdot - 2 - 15 \ln (\frac{| 2 |}{| 1 |})}{3}$

Step 10.3.2

Multiply $4 \cdot 2^{\frac{5}{2}}$ .

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Step 10.3.2.1

Rewrite $4$ as $2^{2}$ .

$\frac{2^{2} \cdot 2^{\frac{5}{2}} + 4 \cdot - 2 - 15 \ln (\frac{| 2 |}{| 1 |})}{3}$

Step 10.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2^{2 + \frac{5}{2}} + 4 \cdot - 2 - 15 \ln (\frac{| 2 |}{| 1 |})}{3}$

Step 10.3.2.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2^{2 \cdot \frac{2}{2} + \frac{5}{2}} + 4 \cdot - 2 - 15 \ln (\frac{| 2 |}{| 1 |})}{3}$

Step 10.3.2.4

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

$\frac{2^{\frac{2 \cdot 2}{2} + \frac{5}{2}} + 4 \cdot - 2 - 15 \ln (\frac{| 2 |}{| 1 |})}{3}$

Step 10.3.2.5

Combine the numerators over the common denominator.

$\frac{2^{\frac{2 \cdot 2 + 5}{2}} + 4 \cdot - 2 - 15 \ln (\frac{| 2 |}{| 1 |})}{3}$

Step 10.3.2.6

Simplify the numerator.

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Step 10.3.2.6.1

Multiply $2$ by $2$ .

$\frac{2^{\frac{4 + 5}{2}} + 4 \cdot - 2 - 15 \ln (\frac{| 2 |}{| 1 |})}{3}$

Step 10.3.2.6.2

Add $4$ and $5$ .

$\frac{2^{\frac{9}{2}} + 4 \cdot - 2 - 15 \ln (\frac{| 2 |}{| 1 |})}{3}$

Step 10.3.3

Multiply $4$ by $- 2$ .

$\frac{2^{\frac{9}{2}} - 8 - 15 \ln (\frac{| 2 |}{| 1 |})}{3}$

Step 10.3.4

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

$\frac{2^{\frac{9}{2}} - 8 - 15 \ln (\frac{2}{| 1 |})}{3}$

Step 10.3.5

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2^{\frac{9}{2}} - 8 - 15 \ln (\frac{2}{1})}{3}$

Step 10.3.6

Divide $2$ by $1$ .

$\frac{2^{\frac{9}{2}} - 8 - 15 \ln (2)}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{2^{\frac{9}{2}} - 8 - 15 \ln (2)}{3}$

Decimal Form:

$1.41006976 \dots$

Step 12