Calculus Examples

$f (x) = x \sqrt{x + 1}$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int x \sqrt{x + 1} d x$

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

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

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

Step 6

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

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

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Step 6.1.1

Differentiate $x + 1$ .

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

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

$1 + 0$

Step 6.1.5

Add $1$ and $0$ .

$1$

Step 6.2

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

Simplify.

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.2.3

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

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

Step 8.2.4

Multiply $2$ by $2$ .

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

Step 8.2.5

Multiply $5$ by $3$ .

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

Step 8.2.6

To write $- \frac{4}{15} u^{\frac{5}{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 8.2.7

Combine $- \frac{4}{15} u^{\frac{5}{2}}$ and $\frac{3}{3}$ .

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

Step 8.2.8

Combine the numerators over the common denominator.

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

Step 8.2.9

Combine $u^{\frac{5}{2}}$ and $\frac{4}{15}$ .

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

Step 8.2.10

Multiply $3$ by $- 1$ .

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

Step 8.2.11

Combine $- 3$ and $\frac{u^{\frac{5}{2}} \cdot 4}{15}$ .

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

Step 8.2.12

Multiply $4$ by $- 3$ .

$\frac{2 x {(x + 1)}^{\frac{3}{2}} + \frac{- 12 u^{\frac{5}{2}}}{15}}{3} + C$

Step 8.2.13

Factor $3$ out of $- 12 u^{\frac{5}{2}}$ .

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

Step 8.2.14

Cancel the common factors.

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Step 8.2.14.1

Factor $3$ out of $15$ .

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

Step 8.2.14.2

Cancel the common factor.

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

Step 8.2.14.3

Rewrite the expression.

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

Step 8.2.15

Move the negative in front of the fraction.

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

Step 8.2.16

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

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

Step 8.2.17

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

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

Step 8.2.18

Combine the numerators over the common denominator.

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

Step 8.2.19

Multiply $5$ by $2$ .

$\frac{\frac{10 x {(x + 1)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{5}}{3} + C$

Step 8.2.20

Rewrite $\frac{\frac{10 x {(x + 1)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{5}}{3}$ as a product.

$\frac{10 x {(x + 1)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{5} \cdot \frac{1}{3} + C$

Step 8.2.21

Multiply $\frac{10 x {(x + 1)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{5}$ by $\frac{1}{3}$ .

$\frac{10 x {(x + 1)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{5 \cdot 3} + C$

Step 8.2.22

Multiply $5$ by $3$ .

$\frac{10 x {(x + 1)}^{\frac{3}{2}} - 4 u^{\frac{5}{2}}}{15} + C$

Step 9

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

$\frac{10 x {(x + 1)}^{\frac{3}{2}} - 4 {(x + 1)}^{\frac{5}{2}}}{15} + C$

Step 10

Reorder terms.

$\frac{1}{15} (10 x {(x + 1)}^{\frac{3}{2}} - 4 {(x + 1)}^{\frac{5}{2}}) + C$

Step 11

The answer is the antiderivative of the function $f (x) = x \sqrt{x + 1}$ .

$F (x) =$ $\frac{1}{15} (10 x {(x + 1)}^{\frac{3}{2}} - 4 {(x + 1)}^{\frac{5}{2}}) + C$