Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{2} x^{2}]_{0}^{1} + \int_{0}^{1} \sqrt{x} d x$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{1}{2} x^{2}]_{0}^{1} + \int_{0}^{1} x^{\frac{1}{2}} d x$

Step 5

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$\frac{1}{2} x^{2}]_{0}^{1} + \frac{2}{3} x^{\frac{3}{2}}]_{0}^{1}$

Step 6

Simplify the answer.

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Step 6.1

Combine $\frac{1}{2} x^{2}]_{0}^{1}$ and $\frac{2}{3} x^{\frac{3}{2}}]_{0}^{1}$ .

$\frac{1}{2} x^{2} + \frac{2}{3} x^{\frac{3}{2}}]_{0}^{1}$

Step 6.2

Substitute and simplify.

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Step 6.2.1

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

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

Step 6.2.2

Simplify.

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Step 6.2.2.1

One to any power is one.

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

Step 6.2.2.2

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

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

Step 6.2.2.3

One to any power is one.

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

Step 6.2.2.4

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

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

Step 6.2.2.5

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

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

Step 6.2.2.6

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.2.2.7

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.2.7.1

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

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

Step 6.2.2.7.2

Multiply $2$ by $3$ .

$\frac{3}{6} + \frac{2}{3} \cdot \frac{2}{2} - (\frac{1}{2} \cdot 0^{2} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.2.7.3

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

$\frac{3}{6} + \frac{2 \cdot 2}{3 \cdot 2} - (\frac{1}{2} \cdot 0^{2} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.2.7.4

Multiply $3$ by $2$ .

$\frac{3}{6} + \frac{2 \cdot 2}{6} - (\frac{1}{2} \cdot 0^{2} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.2.8

Combine the numerators over the common denominator.

$\frac{3 + 2 \cdot 2}{6} - (\frac{1}{2} \cdot 0^{2} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.2.9

Simplify the numerator.

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Step 6.2.2.9.1

Multiply $2$ by $2$ .

$\frac{3 + 4}{6} - (\frac{1}{2} \cdot 0^{2} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.2.9.2

Add $3$ and $4$ .

$\frac{7}{6} - (\frac{1}{2} \cdot 0^{2} + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.2.10

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

$\frac{7}{6} - (\frac{1}{2} \cdot 0 + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.2.11

Multiply $\frac{1}{2}$ by $0$ .

$\frac{7}{6} - (0 + \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.2.12

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

$\frac{7}{6} - (0 + \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}})$

Step 6.2.2.13

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

$\frac{7}{6} - (0 + \frac{2}{3} \cdot 0^{2 (\frac{3}{2})})$

Step 6.2.2.14

Cancel the common factor of $2$ .

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Step 6.2.2.14.1

Cancel the common factor.

$\frac{7}{6} - (0 + \frac{2}{3} \cdot 0^{2 (\frac{3}{2})})$

Step 6.2.2.14.2

Rewrite the expression.

$\frac{7}{6} - (0 + \frac{2}{3} \cdot 0^{3})$

Step 6.2.2.15

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

$\frac{7}{6} - (0 + \frac{2}{3} \cdot 0)$

Step 6.2.2.16

Multiply $\frac{2}{3}$ by $0$ .

$\frac{7}{6} - (0 + 0)$

Step 6.2.2.17

Add $0$ and $0$ .

$\frac{7}{6} - 0$

Step 6.2.2.18

Multiply $- 1$ by $0$ .

$\frac{7}{6} + 0$

Step 6.2.2.19

Add $\frac{7}{6}$ and $0$ .

$\frac{7}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{6}$

Decimal Form:

$1.1\overline{6}$

Mixed Number Form:

$1 \frac{1}{6}$

Step 8