Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Multiply $2$ by $- 1$ .

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

Step 10

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

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

Step 11

Simplify the answer.

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Step 11.1

Substitute and simplify.

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Step 11.1.1

Evaluate $\frac{2 x^{\frac{3}{2}}}{3}$ at $4$ and at $1$ .

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

Step 11.1.2

Evaluate $\ln (| x |)$ at $4$ and at $1$ .

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

Step 11.1.3

Simplify.

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Step 11.1.3.1

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

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

Step 11.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 11.1.3.3

Cancel the common factor of $2$ .

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Step 11.1.3.3.1

Cancel the common factor.

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

Step 11.1.3.3.2

Rewrite the expression.

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

Step 11.1.3.4

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

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

Step 11.1.3.5

Multiply $2$ by $8$ .

$3 (\frac{16}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3}) - 2 (\ln (| 4 |) - \ln (| 1 |))$

Step 11.1.3.6

One to any power is one.

$3 (\frac{16}{3} - \frac{2 \cdot 1}{3}) - 2 (\ln (| 4 |) - \ln (| 1 |))$

Step 11.1.3.7

Multiply $2$ by $1$ .

$3 (\frac{16}{3} - \frac{2}{3}) - 2 (\ln (| 4 |) - \ln (| 1 |))$

Step 11.1.3.8

Combine the numerators over the common denominator.

$3 \frac{16 - 2}{3} - 2 (\ln (| 4 |) - \ln (| 1 |))$

Step 11.1.3.9

Subtract $2$ from $16$ .

$3 (\frac{14}{3}) - 2 (\ln (| 4 |) - \ln (| 1 |))$

Step 11.1.3.10

Combine $3$ and $\frac{14}{3}$ .

$\frac{3 \cdot 14}{3} - 2 (\ln (| 4 |) - \ln (| 1 |))$

Step 11.1.3.11

Multiply $3$ by $14$ .

$\frac{42}{3} - 2 (\ln (| 4 |) - \ln (| 1 |))$

Step 11.1.3.12

Cancel the common factor of $42$ and $3$ .

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Step 11.1.3.12.1

Factor $3$ out of $42$ .

$\frac{3 \cdot 14}{3} - 2 (\ln (| 4 |) - \ln (| 1 |))$

Step 11.1.3.12.2

Cancel the common factors.

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Step 11.1.3.12.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 14}{3 (1)} - 2 (\ln (| 4 |) - \ln (| 1 |))$

Step 11.1.3.12.2.2

Cancel the common factor.

$\frac{3 \cdot 14}{3 \cdot 1} - 2 (\ln (| 4 |) - \ln (| 1 |))$

Step 11.1.3.12.2.3

Rewrite the expression.

$\frac{14}{1} - 2 (\ln (| 4 |) - \ln (| 1 |))$

Step 11.1.3.12.2.4

Divide $14$ by $1$ .

$14 - 2 (\ln (| 4 |) - \ln (| 1 |))$

Step 11.2

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

$14 - 2 \ln (\frac{| 4 |}{| 1 |})$

Step 11.3

Simplify.

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Step 11.3.1

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

$14 - 2 \ln (\frac{4}{| 1 |})$

Step 11.3.2

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

$14 - 2 \ln (\frac{4}{1})$

Step 11.3.3

Divide $4$ by $1$ .

$14 - 2 \ln (4)$

Step 12

Simplify each term.

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Step 12.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$14 - 2 \ln (2^{2})$

Step 12.2

Expand $\ln (2^{2})$ by moving $2$ outside the logarithm.

$14 - 2 (2 \ln (2))$

Step 12.3

Multiply $2$ by $- 2$ .

$14 - 4 \ln (2)$

Step 13

The result can be shown in multiple forms.

Exact Form:

$14 - 4 \ln (2)$

Decimal Form:

$11.22741127 \dots$

Step 14