Calculus Examples

$\int \sqrt{x} (6 - x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 6 - x$ and $d v = \sqrt{x}$ .

$(6 - x) (\frac{2}{3} x^{\frac{3}{2}}) - \int \frac{2}{3} x^{\frac{3}{2}} \cdot - 1 d x$

Step 2

Simplify.

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Step 2.1

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

$(6 - x) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{2}{3} x^{\frac{3}{2}} \cdot - 1 d x$

Step 2.2

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

$(6 - x) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{2 x^{\frac{3}{2}}}{3} \cdot - 1 d x$

Step 2.3

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

$(6 - x) \frac{2 x^{\frac{3}{2}}}{3} - \int \frac{2 x^{\frac{3}{2}} \cdot - 1}{3} d x$

Step 2.4

Multiply $- 1$ by $2$ .

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

Step 2.5

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$(6 - x) \frac{2 x^{\frac{3}{2}}}{3} + 1 \int \frac{2 x^{\frac{3}{2}}}{3} d x$

Step 4.2

Multiply $\int \frac{2 x^{\frac{3}{2}}}{3} d x$ by $1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

Rewrite $(6 - x) \frac{2 x^{\frac{3}{2}}}{3} + \frac{2}{3} (\frac{2}{5} x^{\frac{5}{2}} + C)$ as $4 x^{\frac{3}{2}} - \frac{2 x^{\frac{5}{2}}}{3} + \frac{2}{3} (\frac{2}{5} x^{\frac{5}{2}}) + C$ .

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

Multiply $\frac{2}{3}$ by $\frac{2 x^{\frac{5}{2}}}{5}$ .

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

Step 7.2.3

Multiply $2$ by $2$ .

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

Step 7.2.4

Multiply $3$ by $5$ .

$4 x^{\frac{3}{2}} - \frac{2 x^{\frac{5}{2}}}{3} + \frac{4 x^{\frac{5}{2}}}{15} + C$

Step 7.2.5

To write $- \frac{2 x^{\frac{5}{2}}}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$4 x^{\frac{3}{2}} - \frac{2 x^{\frac{5}{2}}}{3} \cdot \frac{5}{5} + \frac{4 x^{\frac{5}{2}}}{15} + C$

Step 7.2.6

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.6.1

Multiply $\frac{2 x^{\frac{5}{2}}}{3}$ by $\frac{5}{5}$ .

$4 x^{\frac{3}{2}} - \frac{2 x^{\frac{5}{2}} \cdot 5}{3 \cdot 5} + \frac{4 x^{\frac{5}{2}}}{15} + C$

Step 7.2.6.2

Multiply $3$ by $5$ .

$4 x^{\frac{3}{2}} - \frac{2 x^{\frac{5}{2}} \cdot 5}{15} + \frac{4 x^{\frac{5}{2}}}{15} + C$

Step 7.2.7

Combine the numerators over the common denominator.

$4 x^{\frac{3}{2}} + \frac{- 2 x^{\frac{5}{2}} \cdot 5 + 4 x^{\frac{5}{2}}}{15} + C$

Step 7.2.8

Multiply $5$ by $- 2$ .

$4 x^{\frac{3}{2}} + \frac{- 10 x^{\frac{5}{2}} + 4 x^{\frac{5}{2}}}{15} + C$

Step 7.2.9

Add $- 10 x^{\frac{5}{2}}$ and $4 x^{\frac{5}{2}}$ .

$4 x^{\frac{3}{2}} + \frac{- 6 x^{\frac{5}{2}}}{15} + C$

Step 7.2.10

Factor $3$ out of $- 6 x^{\frac{5}{2}}$ .

$4 x^{\frac{3}{2}} + \frac{3 (- 2 x^{\frac{5}{2}})}{15} + C$

Step 7.2.11

Cancel the common factors.

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Step 7.2.11.1

Factor $3$ out of $15$ .

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

Step 7.2.11.2

Cancel the common factor.

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

Step 7.2.11.3

Rewrite the expression.

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

Step 7.2.12

Move the negative in front of the fraction.

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

Step 7.3

Reorder terms.

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