Calculus Examples

$\int x \sqrt{2 x + 1} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sqrt{2 x + 1}$ .

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Let $u = 2 x + 1$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 2 x + 1$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $2 x + 1$ .

$\frac{d}{d x} [2 x + 1]$

Step 4.1.2

By the Sum Rule, the derivative of $2 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [1]$

Step 4.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 4.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

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 $n = 1$ .

$2 \cdot 1 + \frac{d}{d x} [1]$

Step 4.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [1]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 + 0$

Step 4.1.4.2

Add $2$ and $0$ .

$2$

Step 4.2

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

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

Step 5

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

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

Step 6

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Multiply $2$ by $3$ .

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Rewrite $\frac{x {(2 x + 1)}^{\frac{3}{2}}}{3} - \frac{1}{6} (\frac{2}{5} u^{\frac{5}{2}} + C)$ as $\frac{1}{3} x {(2 x + 1)}^{\frac{3}{2}} - \frac{1}{6} \cdot \frac{2}{5} u^{\frac{5}{2}} + C$ .

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

Step 9.2

Simplify.

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.2.3

Multiply $\frac{2}{5}$ by $\frac{1}{6}$ .

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

Step 9.2.4

Multiply $5$ by $6$ .

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

Step 9.2.5

Cancel the common factor of $2$ and $30$ .

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Step 9.2.5.1

Factor $2$ out of $2$ .

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

Step 9.2.5.2

Cancel the common factors.

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Step 9.2.5.2.1

Factor $2$ out of $30$ .

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

Step 9.2.5.2.2

Cancel the common factor.

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

Step 9.2.5.2.3

Rewrite the expression.

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

Step 9.2.6

To write $- \frac{1}{15} u^{\frac{5}{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 9.2.7

Combine $- \frac{1}{15} u^{\frac{5}{2}}$ and $\frac{3}{3}$ .

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

Step 9.2.8

Combine the numerators over the common denominator.

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

Step 9.2.9

Combine $u^{\frac{5}{2}}$ and $\frac{1}{15}$ .

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

Step 9.2.10

Multiply $3$ by $- 1$ .

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

Step 9.2.11

Combine $- 3$ and $\frac{u^{\frac{5}{2}}}{15}$ .

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

Step 9.2.12

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

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

Step 9.2.13

Cancel the common factors.

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Step 9.2.13.1

Factor $3$ out of $15$ .

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

Step 9.2.13.2

Cancel the common factor.

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

Step 9.2.13.3

Rewrite the expression.

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

Step 9.2.14

Move the negative in front of the fraction.

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

Step 9.2.15

To write $x {(2 x + 1)}^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 9.2.16

Combine $x {(2 x + 1)}^{\frac{3}{2}}$ and $\frac{5}{5}$ .

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

Step 9.2.17

Combine the numerators over the common denominator.

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

Step 9.2.18

Move $5$ to the left of $x {(2 x + 1)}^{\frac{3}{2}}$ .

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

Step 9.2.19

Rewrite $\frac{\frac{5 x {(2 x + 1)}^{\frac{3}{2}} - u^{\frac{5}{2}}}{5}}{3}$ as a product.

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

Step 9.2.20

Multiply $\frac{5 x {(2 x + 1)}^{\frac{3}{2}} - u^{\frac{5}{2}}}{5}$ by $\frac{1}{3}$ .

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

Step 9.2.21

Multiply $5$ by $3$ .

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

Step 10

Replace all occurrences of $u$ with $2 x + 1$ .

$\frac{5 x {(2 x + 1)}^{\frac{3}{2}} - {(2 x + 1)}^{\frac{5}{2}}}{15} + C$

Step 11

Reorder terms.

$\frac{1}{15} (5 x {(2 x + 1)}^{\frac{3}{2}} - {(2 x + 1)}^{\frac{5}{2}}) + C$