Calculus Examples

$\int_{0}^{3} x \sqrt{x + 1} d x$

Step 1

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

$x (\frac{2}{3} {(x + 1)}^{\frac{3}{2}})]_{0}^{3} - \int_{0}^{3} \frac{2}{3} {(x + 1)}^{\frac{3}{2}} d x$

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

$x \frac{2 {(x + 1)}^{\frac{3}{2}}}{3}]_{0}^{3} - \int_{0}^{3} \frac{2}{3} {(x + 1)}^{\frac{3}{2}} d x$

Step 2.2

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

$\frac{x (2 {(x + 1)}^{\frac{3}{2}})}{3}]_{0}^{3} - \int_{0}^{3} \frac{2}{3} {(x + 1)}^{\frac{3}{2}} d x$

Step 3

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

$\frac{x (2 {(x + 1)}^{\frac{3}{2}})}{3}]_{0}^{3} - (\frac{2}{3} \int_{0}^{3} {(x + 1)}^{\frac{3}{2}} d x)$

Step 4

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

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

Differentiate $x + 1$ .

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

Step 4.1.2

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 4.1.3

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 4.1.4

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

$1 + 0$

Step 4.1.5

Add $1$ and $0$ .

$1$

Step 4.2

Substitute the lower limit in for $x$ in $u = x + 1$ .

$u_{lower} = 0 + 1$

Step 4.3

Add $0$ and $1$ .

$u_{lower} = 1$

Step 4.4

Substitute the upper limit in for $x$ in $u = x + 1$ .

$u_{upper} = 3 + 1$

Step 4.5

Add $3$ and $1$ .

$u_{upper} = 4$

Step 4.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 4$

Step 4.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\frac{x (2 {(x + 1)}^{\frac{3}{2}})}{3}]_{0}^{3} - \frac{2}{3} \int_{1}^{4} u^{\frac{3}{2}} d u$

Step 5

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

$\frac{x (2 {(x + 1)}^{\frac{3}{2}})}{3}]_{0}^{3} - \frac{2}{3} \frac{2}{5} u^{\frac{5}{2}}]_{1}^{4}$

Step 6

Substitute and simplify.

Tap for more steps...

Step 6.1

Evaluate $\frac{x (2 {(x + 1)}^{\frac{3}{2}})}{3}$ at $3$ and at $0$ .

$(\frac{3 (2 {(3 + 1)}^{\frac{3}{2}})}{3}) - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} \frac{2}{5} u^{\frac{5}{2}}]_{1}^{4}$

Step 6.2

Evaluate $\frac{2}{5} u^{\frac{5}{2}}$ at $4$ and at $1$ .

$(\frac{3 (2 {(3 + 1)}^{\frac{3}{2}})}{3}) - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3

Simplify.

Tap for more steps...

Step 6.3.1

Add $3$ and $1$ .

$\frac{3 (2 \cdot 4^{\frac{3}{2}})}{3} - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.2

Rewrite $4$ as $2^{2}$ .

$\frac{3 (2 \cdot {(2^{2})}^{\frac{3}{2}})}{3} - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.3

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

$\frac{3 (2 \cdot 2^{2 (\frac{3}{2})})}{3} - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.4.1

Cancel the common factor.

$\frac{3 (2 \cdot 2^{2 (\frac{3}{2})})}{3} - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.4.2

Rewrite the expression.

$\frac{3 (2 \cdot 2^{3})}{3} - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.5

Raise $2$ to the power of $3$ .

$\frac{3 (2 \cdot 8)}{3} - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.6

Multiply $2$ by $8$ .

$\frac{3 \cdot 16}{3} - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.7

Multiply $3$ by $16$ .

$\frac{48}{3} - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.8

Cancel the common factor of $48$ and $3$ .

Tap for more steps...

Step 6.3.8.1

Factor $3$ out of $48$ .

$\frac{3 \cdot 16}{3} - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.8.2

Cancel the common factors.

Tap for more steps...

Step 6.3.8.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 16}{3 (1)} - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.8.2.2

Cancel the common factor.

$\frac{3 \cdot 16}{3 \cdot 1} - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.8.2.3

Rewrite the expression.

$\frac{16}{1} - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.8.2.4

Divide $16$ by $1$ .

$16 - \frac{0 (2 {(0 + 1)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.9

Add $0$ and $1$ .

$16 - \frac{0 (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.10

One to any power is one.

$16 - \frac{0 (2 \cdot 1)}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.11

Multiply $2$ by $1$ .

$16 - \frac{0 \cdot 2}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.12

Multiply $0$ by $2$ .

$16 - \frac{0}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.13

Cancel the common factor of $0$ and $3$ .

Tap for more steps...

Step 6.3.13.1

Factor $3$ out of $0$ .

$16 - \frac{3 (0)}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.13.2

Cancel the common factors.

Tap for more steps...

Step 6.3.13.2.1

Factor $3$ out of $3$ .

$16 - \frac{3 \cdot 0}{3 \cdot 1} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.13.2.2

Cancel the common factor.

$16 - \frac{3 \cdot 0}{3 \cdot 1} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.13.2.3

Rewrite the expression.

$16 - \frac{0}{1} - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.13.2.4

Divide $0$ by $1$ .

$16 - 0 - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.14

Multiply $- 1$ by $0$ .

$16 + 0 - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.15

Add $16$ and $0$ .

$16 - \frac{2}{3} ((\frac{2}{5} \cdot 4^{\frac{5}{2}}) - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.16

Rewrite $4$ as $2^{2}$ .

$16 - \frac{2}{3} (\frac{2}{5} \cdot {(2^{2})}^{\frac{5}{2}} - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.17

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

$16 - \frac{2}{3} (\frac{2}{5} \cdot 2^{2 (\frac{5}{2})} - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.18

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.18.1

Cancel the common factor.

$16 - \frac{2}{3} (\frac{2}{5} \cdot 2^{2 (\frac{5}{2})} - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.18.2

Rewrite the expression.

$16 - \frac{2}{3} (\frac{2}{5} \cdot 2^{5} - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.19

Raise $2$ to the power of $5$ .

$16 - \frac{2}{3} (\frac{2}{5} \cdot 32 - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.20

Combine $\frac{2}{5}$ and $32$ .

$16 - \frac{2}{3} (\frac{2 \cdot 32}{5} - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.21

Multiply $2$ by $32$ .

$16 - \frac{2}{3} (\frac{64}{5} - \frac{2}{5} \cdot 1^{\frac{5}{2}})$

Step 6.3.22

One to any power is one.

$16 - \frac{2}{3} (\frac{64}{5} - \frac{2}{5} \cdot 1)$

Step 6.3.23

Multiply $- 1$ by $1$ .

$16 - \frac{2}{3} (\frac{64}{5} - \frac{2}{5})$

Step 6.3.24

Combine the numerators over the common denominator.

$16 - \frac{2}{3} \cdot \frac{64 - 2}{5}$

Step 6.3.25

Subtract $2$ from $64$ .

$16 - \frac{2}{3} \cdot \frac{62}{5}$

Step 6.3.26

Multiply $\frac{62}{5}$ by $\frac{2}{3}$ .

$16 - \frac{62 \cdot 2}{5 \cdot 3}$

Step 6.3.27

Multiply $62$ by $2$ .

$16 - \frac{124}{5 \cdot 3}$

Step 6.3.28

Multiply $5$ by $3$ .

$16 - \frac{124}{15}$

Step 6.3.29

To write $16$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$16 \cdot \frac{15}{15} - \frac{124}{15}$

Step 6.3.30

Combine $16$ and $\frac{15}{15}$ .

$\frac{16 \cdot 15}{15} - \frac{124}{15}$

Step 6.3.31

Combine the numerators over the common denominator.

$\frac{16 \cdot 15 - 124}{15}$

Step 6.3.32

Simplify the numerator.

Tap for more steps...

Step 6.3.32.1

Multiply $16$ by $15$ .

$\frac{240 - 124}{15}$

Step 6.3.32.2

Subtract $124$ from $240$ .

$\frac{116}{15}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{116}{15}$

Decimal Form:

$7.7\overline{3}$

Mixed Number Form:

$7 \frac{11}{15}$

Step 8