Calculus Examples

$\int_{- 4}^{4} 3 \sqrt{x} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{- 4}^{4} \sqrt{x} d x$

Step 2

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

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

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

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

Step 4.2

Substitute and simplify.

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Step 4.2.1

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

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

Cancel the common factor of $2$ .

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Step 4.2.2.3.1

Cancel the common factor.

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

Step 4.2.2.3.2

Rewrite the expression.

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

Step 4.2.2.4

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

$3 (\frac{2 \cdot 8}{3} - \frac{2 {(- 4)}^{\frac{3}{2}}}{3})$

Step 4.2.2.5

Multiply $2$ by $8$ .

$3 (\frac{16}{3} - \frac{2 {(- 4)}^{\frac{3}{2}}}{3})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$3 (\frac{16}{3}) + 3 (- \frac{2 {(- 4)}^{\frac{3}{2}}}{3})$

Step 5.2

Cancel the common factor of $3$ .

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Step 5.2.1

Cancel the common factor.

$3 (\frac{16}{3}) + 3 (- \frac{2 {(- 4)}^{\frac{3}{2}}}{3})$

Step 5.2.2

Rewrite the expression.

$16 + 3 (- \frac{2 {(- 4)}^{\frac{3}{2}}}{3})$

Step 5.3

Cancel the common factor of $3$ .

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Step 5.3.1

Move the leading negative in $- \frac{2 {(- 4)}^{\frac{3}{2}}}{3}$ into the numerator.

$16 + 3 \frac{- 2 {(- 4)}^{\frac{3}{2}}}{3}$

Step 5.3.2

Cancel the common factor.

$16 + 3 \frac{- 2 {(- 4)}^{\frac{3}{2}}}{3}$

Step 5.3.3

Rewrite the expression.

$16 - 2 {(- 4)}^{\frac{3}{2}}$

Step 6