Calculus Examples

$\int \sqrt{x} (3 x - 2) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 3 x - 2$ and $d v = \sqrt{x}$ .

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $3$ by $2$ .

$(3 x - 2) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{6 x^{\frac{3}{2}}}{3} d x$

Step 2.5

Factor $3$ out of $6 x^{\frac{3}{2}}$ .

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

Step 2.6

Cancel the common factors.

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Step 2.6.1

Factor $3$ out of $3$ .

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

Step 2.6.2

Cancel the common factor.

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

Step 2.6.3

Rewrite the expression.

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

Step 2.6.4

Divide $2 x^{\frac{3}{2}}$ by $1$ .

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

Step 3

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

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

Step 4

Multiply $2$ by $- 1$ .

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

Step 5

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$(3 x - 2) \frac{2 x^{\frac{3}{2}}}{3} - 2 (\frac{2}{5} x^{\frac{5}{2}} + C)$

Step 6

Simplify the answer.

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Step 6.1

Combine $\frac{2}{5}$ and $x^{\frac{5}{2}}$ .

$(3 x - 2) \frac{2 x^{\frac{3}{2}}}{3} - 2 (\frac{2 x^{\frac{5}{2}}}{5} + C)$

Step 6.2

Rewrite $(3 x - 2) \frac{2 x^{\frac{3}{2}}}{3} - 2 (\frac{2 x^{\frac{5}{2}}}{5} + C)$ as $2 x^{\frac{5}{2}} + \frac{- 4 x^{\frac{3}{2}}}{3} - 2 \frac{2 x^{\frac{5}{2}}}{5} + C$ .

$2 x^{\frac{5}{2}} + \frac{- 4 x^{\frac{3}{2}}}{3} - 2 \frac{2 x^{\frac{5}{2}}}{5} + C$

Step 6.3

Simplify.

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Step 6.3.1

Move the negative in front of the fraction.

$2 x^{\frac{5}{2}} - \frac{4 x^{\frac{3}{2}}}{3} - 2 \frac{2 x^{\frac{5}{2}}}{5} + C$

Step 6.3.2

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

$2 x^{\frac{5}{2}} - \frac{4 x^{\frac{3}{2}}}{3} + \frac{- 2 (2 x^{\frac{5}{2}})}{5} + C$

Step 6.3.3

Multiply $2$ by $- 2$ .

$2 x^{\frac{5}{2}} - \frac{4 x^{\frac{3}{2}}}{3} + \frac{- 4 x^{\frac{5}{2}}}{5} + C$

Step 6.3.4

Move the negative in front of the fraction.

$2 x^{\frac{5}{2}} - \frac{4 x^{\frac{3}{2}}}{3} - \frac{4 x^{\frac{5}{2}}}{5} + C$

Step 6.3.5

To write $2 x^{\frac{5}{2}}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$2 x^{\frac{5}{2}} \cdot \frac{5}{5} - \frac{4 x^{\frac{5}{2}}}{5} - \frac{4 x^{\frac{3}{2}}}{3} + C$

Step 6.3.6

Combine $2 x^{\frac{5}{2}}$ and $\frac{5}{5}$ .

$\frac{2 x^{\frac{5}{2}} \cdot 5}{5} - \frac{4 x^{\frac{5}{2}}}{5} - \frac{4 x^{\frac{3}{2}}}{3} + C$

Step 6.3.7

Combine the numerators over the common denominator.

$\frac{2 x^{\frac{5}{2}} \cdot 5 - 4 x^{\frac{5}{2}}}{5} - \frac{4 x^{\frac{3}{2}}}{3} + C$

Step 6.3.8

Multiply $5$ by $2$ .

$\frac{10 x^{\frac{5}{2}} - 4 x^{\frac{5}{2}}}{5} - \frac{4 x^{\frac{3}{2}}}{3} + C$

Step 6.3.9

Subtract $4 x^{\frac{5}{2}}$ from $10 x^{\frac{5}{2}}$ .

$\frac{6 x^{\frac{5}{2}}}{5} - \frac{4 x^{\frac{3}{2}}}{3} + C$

Step 6.4

Reorder terms.

$\frac{6}{5} x^{\frac{5}{2}} - \frac{4}{3} x^{\frac{3}{2}} + C$