Calculus Examples

$\int \sqrt{x} \ln (2 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (2 x)$ and $d v = \sqrt{x}$ .

$\ln (2 x) (\frac{2}{3} x^{\frac{3}{2}}) - \int \frac{2}{3} x^{\frac{3}{2}} \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

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

$\ln (2 x) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{2}{3} x^{\frac{3}{2}} \frac{1}{x} d x$

Step 2.2

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

$\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - \int \frac{2}{3} x^{\frac{3}{2}} \frac{1}{x} d x$

Step 3

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

$\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - (\frac{2}{3} \int x^{\frac{3}{2}} \frac{1}{x} d x)$

Step 4

Simplify.

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Step 4.1

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

$\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - (\frac{2}{3} \int \frac{x^{\frac{3}{2}}}{x} d x)$

Step 4.2

Move $x^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - (\frac{2}{3} \int x^{\frac{3}{2}} x^{- 1} d x)$

Step 4.3

Multiply $x^{\frac{3}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 4.3.1

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

$\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - (\frac{2}{3} \int x^{\frac{3}{2} - 1} d x)$

Step 4.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - (\frac{2}{3} \int x^{\frac{3}{2} - 1 \cdot \frac{2}{2}} d x)$

Step 4.3.3

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

$\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - (\frac{2}{3} \int x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} d x)$

Step 4.3.4

Combine the numerators over the common denominator.

$\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - (\frac{2}{3} \int x^{\frac{3 - 1 \cdot 2}{2}} d x)$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $- 1$ by $2$ .

$\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - (\frac{2}{3} \int x^{\frac{3 - 2}{2}} d x)$

Step 4.3.5.2

Subtract $2$ from $3$ .

$\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - (\frac{2}{3} \int x^{\frac{1}{2}} d x)$

$\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - \frac{2}{3} \int x^{\frac{1}{2}} d x$

Step 5

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

$\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - \frac{2}{3} (\frac{2}{3} x^{\frac{3}{2}} + C)$

Step 6

Simplify the answer.

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Step 6.1

Rewrite $\frac{\ln (2 x) (2 x^{\frac{3}{2}})}{3} - \frac{2}{3} (\frac{2}{3} x^{\frac{3}{2}} + C)$ as $\frac{2}{3} \ln (2 x) x^{\frac{3}{2}} - \frac{2}{3} \cdot \frac{2}{3} x^{\frac{3}{2}} + C$ .

$\frac{2}{3} \ln (2 x) x^{\frac{3}{2}} - \frac{2}{3} \cdot \frac{2}{3} x^{\frac{3}{2}} + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine $\frac{2}{3}$ and $\ln (2 x)$ .

$\frac{2 \ln (2 x)}{3} x^{\frac{3}{2}} - \frac{2}{3} \cdot \frac{2}{3} x^{\frac{3}{2}} + C$

Step 6.2.2

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

$\frac{2 \ln (2 x) x^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot \frac{2}{3} x^{\frac{3}{2}} + C$

Step 6.2.3

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

$\frac{2 \ln (2 x) x^{\frac{3}{2}}}{3} - \frac{2 \cdot 2}{3 \cdot 3} x^{\frac{3}{2}} + C$

Step 6.2.4

Multiply $2$ by $2$ .

$\frac{2 \ln (2 x) x^{\frac{3}{2}}}{3} - \frac{4}{3 \cdot 3} x^{\frac{3}{2}} + C$

Step 6.2.5

Multiply $3$ by $3$ .

$\frac{2 \ln (2 x) x^{\frac{3}{2}}}{3} - \frac{4}{9} x^{\frac{3}{2}} + C$

$\frac{2}{3} \ln (2 x) x^{\frac{3}{2}} - \frac{4}{9} x^{\frac{3}{2}} + C$