Calculus Examples

$\int (3 - 2 x) \sqrt{9 x} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 3 - 2 x$ and $d v = \sqrt{9 x}$ .

$(3 - 2 x) (\frac{2}{27} {(9 x)}^{\frac{3}{2}}) - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2

Simplify.

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Step 2.1

Factor $9$ out of $9 x$ .

$(3 - 2 x) (\frac{2}{27} {(9 (x))}^{\frac{3}{2}}) - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.2

Apply the product rule to $9 (x)$ .

$(3 - 2 x) (\frac{2}{27} (9^{\frac{3}{2}} x^{\frac{3}{2}})) - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.3

Rewrite $9$ as $3^{2}$ .

$(3 - 2 x) (\frac{2}{27} ({(3^{2})}^{\frac{3}{2}} x^{\frac{3}{2}})) - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.4

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

$(3 - 2 x) (\frac{2}{27} (3^{2 (\frac{3}{2})} x^{\frac{3}{2}})) - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.5

Cancel the common factor of $2$ .

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Step 2.5.1

Cancel the common factor.

$(3 - 2 x) (\frac{2}{27} (3^{2 (\frac{3}{2})} x^{\frac{3}{2}})) - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.5.2

Rewrite the expression.

$(3 - 2 x) (\frac{2}{27} (3^{3} x^{\frac{3}{2}})) - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.6

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

$(3 - 2 x) (\frac{2}{27} (27 x^{\frac{3}{2}})) - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.7

Combine $27$ and $\frac{2}{27}$ .

$(3 - 2 x) (\frac{27 \cdot 2}{27} x^{\frac{3}{2}}) - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.8

Multiply $27$ by $2$ .

$(3 - 2 x) (\frac{54}{27} x^{\frac{3}{2}}) - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.9

Combine $\frac{54}{27}$ and $x^{\frac{3}{2}}$ .

$(3 - 2 x) \frac{54 x^{\frac{3}{2}}}{27} - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.10

Factor $27$ out of $54 x^{\frac{3}{2}}$ .

$(3 - 2 x) \frac{27 (2 x^{\frac{3}{2}})}{27} - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.11

Cancel the common factors.

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Step 2.11.1

Factor $27$ out of $27$ .

$(3 - 2 x) \frac{27 (2 x^{\frac{3}{2}})}{27 (1)} - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.11.2

Cancel the common factor.

$(3 - 2 x) \frac{27 (2 x^{\frac{3}{2}})}{27 \cdot 1} - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.11.3

Rewrite the expression.

$(3 - 2 x) \frac{2 x^{\frac{3}{2}}}{1} - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.11.4

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

$(3 - 2 x) (2 x^{\frac{3}{2}}) - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 2.12

Move $2$ to the left of $3 - 2 x$ .

$2 (3 - 2 x) x^{\frac{3}{2}} - \int \frac{2}{27} {(9 x)}^{\frac{3}{2}} \cdot - 2 d x$

Step 3

Since $\frac{2}{27} \cdot - 2$ is constant with respect to $x$ , move $\frac{2}{27} \cdot - 2$ out of the integral.

$2 (3 - 2 x) x^{\frac{3}{2}} - (\frac{2}{27} \cdot - 2 \int {(9 x)}^{\frac{3}{2}} d x)$

Step 4

Simplify.

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Step 4.1

Combine $\frac{2}{27}$ and $- 2$ .

$2 (3 - 2 x) x^{\frac{3}{2}} - (\frac{2 \cdot - 2}{27} \int {(9 x)}^{\frac{3}{2}} d x)$

Step 4.2

Multiply $2$ by $- 2$ .

$2 (3 - 2 x) x^{\frac{3}{2}} - (\frac{- 4}{27} \int {(9 x)}^{\frac{3}{2}} d x)$

Step 4.3

Move the negative in front of the fraction.

$2 (3 - 2 x) x^{\frac{3}{2}} - (- \frac{4}{27} \int {(9 x)}^{\frac{3}{2}} d x)$

Step 4.4

Factor $9$ out of $9 x$ .

$2 (3 - 2 x) x^{\frac{3}{2}} - (- \frac{4}{27} \int {(9 (x))}^{\frac{3}{2}} d x)$

Step 4.5

Apply the product rule to $9 (x)$ .

$2 (3 - 2 x) x^{\frac{3}{2}} - (- \frac{4}{27} \int 9^{\frac{3}{2}} x^{\frac{3}{2}} d x)$

Step 4.6

Rewrite $9$ as $3^{2}$ .

$2 (3 - 2 x) x^{\frac{3}{2}} - (- \frac{4}{27} \int {(3^{2})}^{\frac{3}{2}} x^{\frac{3}{2}} d x)$

Step 4.7

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

$2 (3 - 2 x) x^{\frac{3}{2}} - (- \frac{4}{27} \int 3^{2 (\frac{3}{2})} x^{\frac{3}{2}} d x)$

Step 4.8

Cancel the common factor of $2$ .

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Step 4.8.1

Cancel the common factor.

$2 (3 - 2 x) x^{\frac{3}{2}} - (- \frac{4}{27} \int 3^{2 (\frac{3}{2})} x^{\frac{3}{2}} d x)$

Step 4.8.2

Rewrite the expression.

$2 (3 - 2 x) x^{\frac{3}{2}} - (- \frac{4}{27} \int 3^{3} x^{\frac{3}{2}} d x)$

Step 4.9

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

$2 (3 - 2 x) x^{\frac{3}{2}} - (- \frac{4}{27} \int 27 x^{\frac{3}{2}} d x)$

Step 4.10

Multiply $- 1$ by $- 1$ .

$2 (3 - 2 x) x^{\frac{3}{2}} + 1 (\frac{4}{27} \int 27 x^{\frac{3}{2}} d x)$

Step 4.11

Multiply $\frac{4}{27}$ by $1$ .

$2 (3 - 2 x) x^{\frac{3}{2}} + \frac{4}{27} \int 27 x^{\frac{3}{2}} d x$

Step 5

Since $27$ is constant with respect to $x$ , move $27$ out of the integral.

$2 (3 - 2 x) x^{\frac{3}{2}} + \frac{4}{27} (27 \int x^{\frac{3}{2}} d x)$

Step 6

Simplify.

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Step 6.1

Combine $27$ and $\frac{4}{27}$ .

$2 (3 - 2 x) x^{\frac{3}{2}} + \frac{27 \cdot 4}{27} \int x^{\frac{3}{2}} d x$

Step 6.2

Multiply $27$ by $4$ .

$2 (3 - 2 x) x^{\frac{3}{2}} + \frac{108}{27} \int x^{\frac{3}{2}} d x$

Step 6.3

Cancel the common factor of $108$ and $27$ .

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Step 6.3.1

Factor $27$ out of $108$ .

$2 (3 - 2 x) x^{\frac{3}{2}} + \frac{27 \cdot 4}{27} \int x^{\frac{3}{2}} d x$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $27$ out of $27$ .

$2 (3 - 2 x) x^{\frac{3}{2}} + \frac{27 \cdot 4}{27 (1)} \int x^{\frac{3}{2}} d x$

Step 6.3.2.2

Cancel the common factor.

$2 (3 - 2 x) x^{\frac{3}{2}} + \frac{27 \cdot 4}{27 \cdot 1} \int x^{\frac{3}{2}} d x$

Step 6.3.2.3

Rewrite the expression.

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

Step 6.3.2.4

Divide $4$ by $1$ .

$2 (3 - 2 x) x^{\frac{3}{2}} + 4 \int x^{\frac{3}{2}} d x$

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

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

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

Step 8.2

Simplify.

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.2.3

Multiply $2$ by $4$ .

$(6 - 4 x) x^{\frac{3}{2}} + \frac{8 x^{\frac{5}{2}}}{5} + C$

Step 8.2.4

To write $(6 - 4 x) x^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$(6 - 4 x) x^{\frac{3}{2}} \cdot \frac{5}{5} + \frac{8 x^{\frac{5}{2}}}{5} + C$

Step 8.2.5

Combine $(6 - 4 x) x^{\frac{3}{2}}$ and $\frac{5}{5}$ .

$\frac{(6 - 4 x) x^{\frac{3}{2}} \cdot 5}{5} + \frac{8 x^{\frac{5}{2}}}{5} + C$

Step 8.2.6

Combine the numerators over the common denominator.

$\frac{(6 - 4 x) x^{\frac{3}{2}} \cdot 5 + 8 x^{\frac{5}{2}}}{5} + C$

Step 8.2.7

Move $5$ to the left of $(6 - 4 x) x^{\frac{3}{2}}$ .

$\frac{5 (6 - 4 x) x^{\frac{3}{2}} + 8 x^{\frac{5}{2}}}{5} + C$

Step 8.3

Reorder terms.

$\frac{1}{5} (5 (6 - 4 x) x^{\frac{3}{2}} + 8 x^{\frac{5}{2}}) + C$