Calculus Examples

$\int_{1}^{4} x^{2} + \sqrt{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{4} x^{2} d x + \int_{1}^{4} \sqrt{x} d x$

Step 2

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

$\frac{1}{3} x^{3}]_{1}^{4} + \int_{1}^{4} \sqrt{x} d x$

Step 3

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

$\frac{1}{3} x^{3}]_{1}^{4} + \int_{1}^{4} x^{\frac{1}{2}} d x$

Step 4

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

$\frac{1}{3} x^{3}]_{1}^{4} + \frac{2}{3} x^{\frac{3}{2}}]_{1}^{4}$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{3} x^{3}]_{1}^{4}$ and $\frac{2}{3} x^{\frac{3}{2}}]_{1}^{4}$ .

$\frac{1}{3} x^{3} + \frac{2}{3} x^{\frac{3}{2}}]_{1}^{4}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{1}{3} x^{3} + \frac{2}{3} x^{\frac{3}{2}}$ at $4$ and at $1$ .

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

Raise $4$ to the power of $3$ .

$\frac{1}{3} \cdot 64 + \frac{2}{3} \cdot 4^{\frac{3}{2}} - (\frac{1}{3} \cdot 1^{3} + \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.2.2

Combine $\frac{1}{3}$ and $64$ .

$\frac{64}{3} + \frac{2}{3} \cdot 4^{\frac{3}{2}} - (\frac{1}{3} \cdot 1^{3} + \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.2.3

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

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

Step 5.2.2.4

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

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

Step 5.2.2.5

Cancel the common factor of $2$ .

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Step 5.2.2.5.1

Cancel the common factor.

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

Step 5.2.2.5.2

Rewrite the expression.

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

Step 5.2.2.6

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

$\frac{64}{3} + \frac{2}{3} \cdot 8 - (\frac{1}{3} \cdot 1^{3} + \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.2.7

Combine $\frac{2}{3}$ and $8$ .

$\frac{64}{3} + \frac{2 \cdot 8}{3} - (\frac{1}{3} \cdot 1^{3} + \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.2.8

Multiply $2$ by $8$ .

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

Step 5.2.2.9

Combine the numerators over the common denominator.

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

Step 5.2.2.10

Add $64$ and $16$ .

$\frac{80}{3} - (\frac{1}{3} \cdot 1^{3} + \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.2.11

One to any power is one.

$\frac{80}{3} - (\frac{1}{3} \cdot 1 + \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.2.12

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

$\frac{80}{3} - (\frac{1}{3} + \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.2.13

One to any power is one.

$\frac{80}{3} - (\frac{1}{3} + \frac{2}{3} \cdot 1)$

Step 5.2.2.14

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

$\frac{80}{3} - (\frac{1}{3} + \frac{2}{3})$

Step 5.2.2.15

Combine the numerators over the common denominator.

$\frac{80}{3} - \frac{1 + 2}{3}$

Step 5.2.2.16

Add $1$ and $2$ .

$\frac{80}{3} - \frac{3}{3}$

Step 5.2.2.17

Cancel the common factor of $3$ .

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Step 5.2.2.17.1

Cancel the common factor.

$\frac{80}{3} - \frac{3}{3}$

Step 5.2.2.17.2

Rewrite the expression.

$\frac{80}{3} - 1 \cdot 1$

Step 5.2.2.18

Multiply $- 1$ by $1$ .

$\frac{80}{3} - 1$

Step 5.2.2.19

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

$\frac{80}{3} - 1 \cdot \frac{3}{3}$

Step 5.2.2.20

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

$\frac{80}{3} + \frac{- 1 \cdot 3}{3}$

Step 5.2.2.21

Combine the numerators over the common denominator.

$\frac{80 - 1 \cdot 3}{3}$

Step 5.2.2.22

Simplify the numerator.

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Step 5.2.2.22.1

Multiply $- 1$ by $3$ .

$\frac{80 - 3}{3}$

Step 5.2.2.22.2

Subtract $3$ from $80$ .

$\frac{77}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{77}{3}$

Decimal Form:

$25.\overline{6}$

Mixed Number Form:

$25 \frac{2}{3}$

Step 7