Calculus Examples

$\int x \sqrt{3 x - 4} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sqrt{3 x - 4}$ .

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

Step 2

Simplify.

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Step 2.1

Combine $\frac{2}{9}$ and ${(3 x - 4)}^{\frac{3}{2}}$ .

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

Step 2.2

Combine $x$ and $\frac{2 {(3 x - 4)}^{\frac{3}{2}}}{9}$ .

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

Step 3

Since $\frac{2}{9}$ is constant with respect to $x$ , move $\frac{2}{9}$ out of the integral.

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

Step 4

Let $u = 3 x - 4$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 3 x - 4$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $3 x - 4$ .

$\frac{d}{d x} [3 x - 4]$

Step 4.1.2

By the Sum Rule, the derivative of $3 x - 4$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 4]$

Step 4.1.3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 4.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- 4]$

Step 4.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1 + \frac{d}{d x} [- 4]$

Step 4.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- 4]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$3 + 0$

Step 4.1.4.2

Add $3$ and $0$ .

$3$

Step 4.2

Rewrite the problem using $u$ and $d u$ .

$\frac{x (2 {(3 x - 4)}^{\frac{3}{2}})}{9} - \frac{2}{9} \int u^{\frac{3}{2}} \frac{1}{3} d u$

Step 5

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

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

Step 6

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{x (2 {(3 x - 4)}^{\frac{3}{2}})}{9} - \frac{2}{9} (\frac{1}{3} \int u^{\frac{3}{2}} d u)$

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{3}$ by $\frac{2}{9}$ .

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

Step 7.2

Multiply $3$ by $9$ .

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

Step 8

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

$\frac{x (2 {(3 x - 4)}^{\frac{3}{2}})}{9} - \frac{2}{27} (\frac{2}{5} u^{\frac{5}{2}} + C)$

Step 9

Simplify.

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Step 9.1

Rewrite $\frac{x (2 {(3 x - 4)}^{\frac{3}{2}})}{9} - \frac{2}{27} (\frac{2}{5} u^{\frac{5}{2}} + C)$ as $\frac{2}{9} x {(3 x - 4)}^{\frac{3}{2}} - \frac{2}{27} \cdot \frac{2}{5} u^{\frac{5}{2}} + C$ .

$\frac{2}{9} x {(3 x - 4)}^{\frac{3}{2}} - \frac{2}{27} \cdot \frac{2}{5} u^{\frac{5}{2}} + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine $\frac{2}{9}$ and $x$ .

$\frac{2 x}{9} {(3 x - 4)}^{\frac{3}{2}} - \frac{2}{27} \cdot \frac{2}{5} u^{\frac{5}{2}} + C$

Step 9.2.2

Combine $\frac{2 x}{9}$ and ${(3 x - 4)}^{\frac{3}{2}}$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}}}{9} - \frac{2}{27} \cdot \frac{2}{5} u^{\frac{5}{2}} + C$

Step 9.2.3

Multiply $\frac{2}{5}$ by $\frac{2}{27}$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}}}{9} - \frac{2 \cdot 2}{5 \cdot 27} u^{\frac{5}{2}} + C$

Step 9.2.4

Multiply $2$ by $2$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}}}{9} - \frac{4}{5 \cdot 27} u^{\frac{5}{2}} + C$

Step 9.2.5

Multiply $5$ by $27$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}}}{9} - \frac{4}{135} u^{\frac{5}{2}} + C$

Step 9.2.6

To write $- \frac{4}{135} u^{\frac{5}{2}}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}}}{9} - \frac{4}{135} u^{\frac{5}{2}} \cdot \frac{9}{9} + C$

Step 9.2.7

Combine $- \frac{4}{135} u^{\frac{5}{2}}$ and $\frac{9}{9}$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}}}{9} + \frac{- \frac{4}{135} u^{\frac{5}{2}} \cdot 9}{9} + C$

Step 9.2.8

Combine the numerators over the common denominator.

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}} - \frac{4}{135} u^{\frac{5}{2}} \cdot 9}{9} + C$

Step 9.2.9

Combine $u^{\frac{5}{2}}$ and $\frac{4}{135}$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}} - \frac{u^{\frac{5}{2}} \cdot 4}{135} \cdot 9}{9} + C$

Step 9.2.10

Multiply $9$ by $- 1$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}} - 9 \frac{u^{\frac{5}{2}} \cdot 4}{135}}{9} + C$

Step 9.2.11

Combine $- 9$ and $\frac{u^{\frac{5}{2}} \cdot 4}{135}$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}} + \frac{- 9 (u^{\frac{5}{2}} \cdot 4)}{135}}{9} + C$

Step 9.2.12

Multiply $4$ by $- 9$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}} + \frac{- 36 u^{\frac{5}{2}}}{135}}{9} + C$

Step 9.2.13

Factor $9$ out of $- 36 u^{\frac{5}{2}}$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}} + \frac{9 (- 4 u^{\frac{5}{2}})}{135}}{9} + C$

Step 9.2.14

Cancel the common factors.

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Step 9.2.14.1

Factor $9$ out of $135$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}} + \frac{9 (- 4 u^{\frac{5}{2}})}{9 (15)}}{9} + C$

Step 9.2.14.2

Cancel the common factor.

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}} + \frac{9 (- 4 u^{\frac{5}{2}})}{9 \cdot 15}}{9} + C$

Step 9.2.14.3

Rewrite the expression.

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}} + \frac{- 4 u^{\frac{5}{2}}}{15}}{9} + C$

Step 9.2.15

Move the negative in front of the fraction.

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}} - \frac{4 u^{\frac{5}{2}}}{15}}{9} + C$

Step 9.2.16

To write $2 x {(3 x - 4)}^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{2 x {(3 x - 4)}^{\frac{3}{2}} \cdot \frac{15}{15} - \frac{4 u^{\frac{5}{2}}}{15}}{9} + C$

Step 9.2.17

Combine $2 x {(3 x - 4)}^{\frac{3}{2}}$ and $\frac{15}{15}$ .

$\frac{\frac{2 x {(3 x - 4)}^{\frac{3}{2}} \cdot 15}{15} - \frac{4 u^{\frac{5}{2}}}{15}}{9} + C$

Step 9.2.18

Combine the numerators over the common denominator.

$\frac{\frac{2 x {(3 x - 4)}^{\frac{3}{2}} \cdot 15 - 4 u^{\frac{5}{2}}}{15}}{9} + C$

Step 9.2.19

Multiply $15$ by $2$ .

$\frac{\frac{30 x {(3 x - 4)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{15}}{9} + C$

Step 9.2.20

Rewrite $\frac{\frac{30 x {(3 x - 4)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{15}}{9}$ as a product.

$\frac{30 x {(3 x - 4)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{15} \cdot \frac{1}{9} + C$

Step 9.2.21

Multiply $\frac{30 x {(3 x - 4)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{15}$ by $\frac{1}{9}$ .

$\frac{30 x {(3 x - 4)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{15 \cdot 9} + C$

Step 9.2.22

Multiply $15$ by $9$ .

$\frac{30 x {(3 x - 4)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{135} + C$

Step 10

Replace all occurrences of $u$ with $3 x - 4$ .

$\frac{30 x {(3 x - 4)}^{\frac{3}{2}} - 4 {(3 x - 4)}^{\frac{5}{2}}}{135} + C$

Step 11

Reorder terms.

$\frac{1}{135} (30 x {(3 x - 4)}^{\frac{3}{2}} - 4 {(3 x - 4)}^{\frac{5}{2}}) + C$