Calculus Examples

$\int_{1}^{4} (5 x^{2} + \sqrt{x}) d x$

Step 1

Remove parentheses.

$\int_{1}^{4} 5 x^{2} + \sqrt{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{4} 5 x^{2} d x + \int_{1}^{4} \sqrt{x} d x$

Step 3

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{1}^{4} x^{2} d x + \int_{1}^{4} \sqrt{x} d x$

Step 4

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

$5 (\frac{1}{3} x^{3}]_{1}^{4}) + \int_{1}^{4} \sqrt{x} d x$

Step 5

Combine fractions.

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Step 5.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$5 (\frac{x^{3}}{3}]_{1}^{4}) + \int_{1}^{4} \sqrt{x} d x$

Step 5.2

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

$5 (\frac{x^{3}}{3}]_{1}^{4}) + \int_{1}^{4} x^{\frac{1}{2}} d x$

Step 6

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

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

Step 7

Substitute and simplify.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Simplify.

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Step 7.3.1

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

$5 (\frac{64}{3} - \frac{1^{3}}{3}) + \frac{2}{3} \cdot 4^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.2

One to any power is one.

$5 (\frac{64}{3} - \frac{1}{3}) + \frac{2}{3} \cdot 4^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.3

Combine the numerators over the common denominator.

$5 \frac{64 - 1}{3} + \frac{2}{3} \cdot 4^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.4

Subtract $1$ from $64$ .

$5 (\frac{63}{3}) + \frac{2}{3} \cdot 4^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.5

Cancel the common factor of $63$ and $3$ .

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Step 7.3.5.1

Factor $3$ out of $63$ .

$5 \frac{3 \cdot 21}{3} + \frac{2}{3} \cdot 4^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.5.2

Cancel the common factors.

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Step 7.3.5.2.1

Factor $3$ out of $3$ .

$5 \frac{3 \cdot 21}{3 (1)} + \frac{2}{3} \cdot 4^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.5.2.2

Cancel the common factor.

$5 \frac{3 \cdot 21}{3 \cdot 1} + \frac{2}{3} \cdot 4^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.5.2.3

Rewrite the expression.

$5 (\frac{21}{1}) + \frac{2}{3} \cdot 4^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.5.2.4

Divide $21$ by $1$ .

$5 \cdot 21 + \frac{2}{3} \cdot 4^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.6

Multiply $5$ by $21$ .

$105 + \frac{2}{3} \cdot 4^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.7

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

$105 + \frac{2}{3} \cdot {(2^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.8

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

$105 + \frac{2}{3} \cdot 2^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.9

Cancel the common factor of $2$ .

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Step 7.3.9.1

Cancel the common factor.

$105 + \frac{2}{3} \cdot 2^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.9.2

Rewrite the expression.

$105 + \frac{2}{3} \cdot 2^{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.10

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

$105 + \frac{2}{3} \cdot 8 - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.11

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

$105 + \frac{2 \cdot 8}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.12

Multiply $2$ by $8$ .

$105 + \frac{16}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}$

Step 7.3.13

One to any power is one.

$105 + \frac{16}{3} - \frac{2}{3} \cdot 1$

Step 7.3.14

Multiply $- 1$ by $1$ .

$105 + \frac{16}{3} - \frac{2}{3}$

Step 7.3.15

Combine the numerators over the common denominator.

$105 + \frac{16 - 2}{3}$

Step 7.3.16

Subtract $2$ from $16$ .

$105 + \frac{14}{3}$

Step 7.3.17

To write $105$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$105 \cdot \frac{3}{3} + \frac{14}{3}$

Step 7.3.18

Combine $105$ and $\frac{3}{3}$ .

$\frac{105 \cdot 3}{3} + \frac{14}{3}$

Step 7.3.19

Combine the numerators over the common denominator.

$\frac{105 \cdot 3 + 14}{3}$

Step 7.3.20

Simplify the numerator.

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Step 7.3.20.1

Multiply $105$ by $3$ .

$\frac{315 + 14}{3}$

Step 7.3.20.2

Add $315$ and $14$ .

$\frac{329}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{329}{3}$

Decimal Form:

$109.\overline{6}$

Mixed Number Form:

$109 \frac{2}{3}$

Step 9