Calculus Examples

$\int 10 w \sqrt{w} d w$

Step 1

Simplify.

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Step 1.1

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

$\int 10 w \cdot w^{\frac{1}{2}} d w$

Step 1.2

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

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Step 1.2.1

Move $w^{\frac{1}{2}}$ .

$\int 10 (w^{\frac{1}{2}} w) d w$

Step 1.2.2

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

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Step 1.2.2.1

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

$\int 10 (w^{\frac{1}{2}} w^{1}) d w$

Step 1.2.2.2

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

$\int 10 w^{\frac{1}{2} + 1} d w$

Step 1.2.3

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

$\int 10 w^{\frac{1}{2} + \frac{2}{2}} d w$

Step 1.2.4

Combine the numerators over the common denominator.

$\int 10 w^{\frac{1 + 2}{2}} d w$

Step 1.2.5

Add $1$ and $2$ .

$\int 10 w^{\frac{3}{2}} d w$

Step 2

Since $10$ is constant with respect to $w$ , move $10$ out of the integral.

$10 \int w^{\frac{3}{2}} d w$

Step 3

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

$10 (\frac{2}{5} w^{\frac{5}{2}} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $10 (\frac{2}{5} w^{\frac{5}{2}} + C)$ as $10 (\frac{2}{5}) w^{\frac{5}{2}} + C$ .

$10 (\frac{2}{5}) w^{\frac{5}{2}} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $10$ and $\frac{2}{5}$ .

$\frac{10 \cdot 2}{5} w^{\frac{5}{2}} + C$

Step 4.2.2

Multiply $10$ by $2$ .

$\frac{20}{5} w^{\frac{5}{2}} + C$

Step 4.2.3

Cancel the common factor of $20$ and $5$ .

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Step 4.2.3.1

Factor $5$ out of $20$ .

$\frac{5 \cdot 4}{5} w^{\frac{5}{2}} + C$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $5$ out of $5$ .

$\frac{5 \cdot 4}{5 (1)} w^{\frac{5}{2}} + C$

Step 4.2.3.2.2

Cancel the common factor.

$\frac{5 \cdot 4}{5 \cdot 1} w^{\frac{5}{2}} + C$

Step 4.2.3.2.3

Rewrite the expression.

$\frac{4}{1} w^{\frac{5}{2}} + C$

Step 4.2.3.2.4

Divide $4$ by $1$ .

$4 w^{\frac{5}{2}} + C$