Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Substitute and simplify.

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Step 5.1

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

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

Step 5.2

Evaluate $\frac{2}{5} x^{\frac{5}{2}}$ at $1$ and at $0$ .

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

Step 5.3

Simplify.

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Step 5.3.1

Add $1$ and $1$ .

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

Step 5.3.2

One to any power is one.

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

Step 5.3.3

Multiply $2$ by $1$ .

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

Step 5.3.4

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

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

Step 5.3.5

Multiply $2$ by $2$ .

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

Step 5.3.6

Add $0$ and $1$ .

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

Step 5.3.7

Multiply $- 1$ by $1$ .

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

Step 5.3.8

Rewrite $0$ as $0^{2}$ .

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

Step 5.3.9

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

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

Step 5.3.10

Cancel the common factor of $2$ .

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Step 5.3.10.1

Cancel the common factor.

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

Step 5.3.10.2

Rewrite the expression.

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

Step 5.3.11

Raising $0$ to any positive power yields $0$ .

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

Step 5.3.12

Multiply $2$ by $0$ .

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

Step 5.3.13

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

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Step 5.3.13.1

Factor $3$ out of $0$ .

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

Step 5.3.13.2

Cancel the common factors.

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Step 5.3.13.2.1

Factor $3$ out of $3$ .

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

Step 5.3.13.2.2

Cancel the common factor.

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

Step 5.3.13.2.3

Rewrite the expression.

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

Step 5.3.13.2.4

Divide $0$ by $1$ .

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

Step 5.3.14

Multiply $- 1$ by $0$ .

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

Step 5.3.15

Add $\frac{4}{3}$ and $0$ .

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

Step 5.3.16

One to any power is one.

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

Step 5.3.17

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

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

Step 5.3.18

Rewrite $0$ as $0^{2}$ .

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

Step 5.3.19

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

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

Step 5.3.20

Cancel the common factor of $2$ .

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Step 5.3.20.1

Cancel the common factor.

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

Step 5.3.20.2

Rewrite the expression.

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

Step 5.3.21

Raising $0$ to any positive power yields $0$ .

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

Step 5.3.22

Multiply $0$ by $- 1$ .

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

Step 5.3.23

Multiply $0$ by $\frac{2}{5}$ .

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

Step 5.3.24

Add $\frac{2}{5}$ and $0$ .

$\frac{4}{3} - \frac{2}{3} \cdot \frac{2}{5}$

Step 5.3.25

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

$\frac{4}{3} - \frac{2 \cdot 2}{5 \cdot 3}$

Step 5.3.26

Multiply $2$ by $2$ .

$\frac{4}{3} - \frac{4}{5 \cdot 3}$

Step 5.3.27

Multiply $5$ by $3$ .

$\frac{4}{3} - \frac{4}{15}$

Step 5.3.28

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

$\frac{4}{3} \cdot \frac{5}{5} - \frac{4}{15}$

Step 5.3.29

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.29.1

Multiply $\frac{4}{3}$ by $\frac{5}{5}$ .

$\frac{4 \cdot 5}{3 \cdot 5} - \frac{4}{15}$

Step 5.3.29.2

Multiply $3$ by $5$ .

$\frac{4 \cdot 5}{15} - \frac{4}{15}$

Step 5.3.30

Combine the numerators over the common denominator.

$\frac{4 \cdot 5 - 4}{15}$

Step 5.3.31

Simplify the numerator.

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Step 5.3.31.1

Multiply $4$ by $5$ .

$\frac{20 - 4}{15}$

Step 5.3.31.2

Subtract $4$ from $20$ .

$\frac{16}{15}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{16}{15}$

Decimal Form:

$1.0\overline{6}$

Mixed Number Form:

$1 \frac{1}{15}$

Step 7