Calculus Examples

$\int (7 u^{\frac{3}{2}} + 2 u^{\frac{1}{2}}) d u$

Step 1

Remove parentheses.

$\int 7 u^{\frac{3}{2}} + 2 u^{\frac{1}{2}} d u$

Step 2

Split the single integral into multiple integrals.

$\int 7 u^{\frac{3}{2}} d u + \int 2 u^{\frac{1}{2}} d u$

Step 3

Since $7$ is constant with respect to $u$ , move $7$ out of the integral.

$7 \int u^{\frac{3}{2}} d u + \int 2 u^{\frac{1}{2}} d u$

Step 4

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

$7 (\frac{2}{5} u^{\frac{5}{2}} + C) + \int 2 u^{\frac{1}{2}} d u$

Step 5

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$7 (\frac{2}{5} u^{\frac{5}{2}} + C) + 2 \int u^{\frac{1}{2}} d u$

Step 6

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

$7 (\frac{2}{5} u^{\frac{5}{2}} + C) + 2 (\frac{2}{3} u^{\frac{3}{2}} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$7 (\frac{2}{5}) u^{\frac{5}{2}} + 2 (\frac{2}{3}) u^{\frac{3}{2}} + C$

Step 7.2

Simplify.

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Step 7.2.1

Combine $7$ and $\frac{2}{5}$ .

$\frac{7 \cdot 2}{5} u^{\frac{5}{2}} + 2 (\frac{2}{3}) u^{\frac{3}{2}} + C$

Step 7.2.2

Multiply $7$ by $2$ .

$\frac{14}{5} u^{\frac{5}{2}} + 2 (\frac{2}{3}) u^{\frac{3}{2}} + C$

Step 7.2.3

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

$\frac{14}{5} u^{\frac{5}{2}} + \frac{2 \cdot 2}{3} u^{\frac{3}{2}} + C$

Step 7.2.4

Multiply $2$ by $2$ .

$\frac{14}{5} u^{\frac{5}{2}} + \frac{4}{3} u^{\frac{3}{2}} + C$