Calculus Examples

$\int 5 {(\sqrt{v^{2} + 6})}^{2} d v$

Step 1

Rewrite ${(\sqrt{v^{2} + 6})}^{2}$ as $v^{2} + 6$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{v^{2} + 6}$ as ${(v^{2} + 6)}^{\frac{1}{2}}$ .

$\int 5 {({(v^{2} + 6)}^{\frac{1}{2}})}^{2} d v$

Step 1.2

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

$\int 5 {(v^{2} + 6)}^{\frac{1}{2} \cdot 2} d v$

Step 1.3

Combine $\frac{1}{2}$ and $2$ .

$\int 5 {(v^{2} + 6)}^{\frac{2}{2}} d v$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$\int 5 {(v^{2} + 6)}^{\frac{2}{2}} d v$

Step 1.4.2

Rewrite the expression.

$\int 5 {(v^{2} + 6)}^{1} d v$

Step 1.5

Simplify.

$\int 5 (v^{2} + 6) d v$

Step 2

Multiply $5 (v^{2} + 6)$ .

$\int 5 v^{2} + 5 \cdot 6 d v$

Step 3

Multiply $5$ by $6$ .

$\int 5 v^{2} + 30 d v$

Step 4

Split the single integral into multiple integrals.

$\int 5 v^{2} d v + \int 30 d v$

Step 5

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

$5 \int v^{2} d v + \int 30 d v$

Step 6

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

$5 (\frac{1}{3} v^{3} + C) + \int 30 d v$

Step 7

Apply the constant rule.

$5 (\frac{1}{3} v^{3} + C) + 30 v + C$

Step 8

Simplify.

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Step 8.1

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

$5 (\frac{v^{3}}{3} + C) + 30 v + C$

Step 8.2

Simplify.

$\frac{5 v^{3}}{3} + 30 v + C$

Step 8.3

Reorder terms.

$\frac{5}{3} v^{3} + 30 v + C$