Calculus Examples

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

Step 1

Simplify.

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Step 1.1

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

$\int_{1}^{4} x \cdot x^{\frac{1}{2}} d x$

Step 1.2

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 1.2.1

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

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Step 1.2.1.1

Raise $x$ to the power of $1$ .

$\int_{1}^{4} x^{1} x^{\frac{1}{2}} d x$

Step 1.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.2.2

Write $1$ as a fraction with a common denominator.

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.2.4

Add $2$ and $1$ .

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

Step 2

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

$\frac{2}{5} x^{\frac{5}{2}}]_{1}^{4}$

Step 3

Substitute and simplify.

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Step 3.1

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

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

Step 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor of $2$ .

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Step 3.2.3.1

Cancel the common factor.

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

Step 3.2.3.2

Rewrite the expression.

$\frac{2}{5} \cdot 2^{5} - \frac{2}{5} \cdot 1^{\frac{5}{2}}$

Step 3.2.4

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

$\frac{2}{5} \cdot 32 - \frac{2}{5} \cdot 1^{\frac{5}{2}}$

Step 3.2.5

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

$\frac{2 \cdot 32}{5} - \frac{2}{5} \cdot 1^{\frac{5}{2}}$

Step 3.2.6

Multiply $2$ by $32$ .

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

Step 3.2.7

One to any power is one.

$\frac{64}{5} - \frac{2}{5} \cdot 1$

Step 3.2.8

Multiply $- 1$ by $1$ .

$\frac{64}{5} - \frac{2}{5}$

Step 3.2.9

Combine the numerators over the common denominator.

$\frac{64 - 2}{5}$

Step 3.2.10

Subtract $2$ from $64$ .

$\frac{62}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{62}{5}$

Decimal Form:

$12.4$

Mixed Number Form:

$12 \frac{2}{5}$

Step 5