Calculus Examples

$\frac{4}{5} x^{\frac{5}{4}} - \frac{1}{2} x^{2}$

Step 1

Split the single integral into multiple integrals.

$\int \frac{4}{5} x^{\frac{5}{4}} d x + \int - \frac{1}{2} x^{2} d x$

Step 2

Since $\frac{4}{5}$ is constant with respect to $x$ , move $\frac{4}{5}$ out of the integral.

$\frac{4}{5} \int x^{\frac{5}{4}} d x + \int - \frac{1}{2} x^{2} d x$

Step 3

By the Power Rule, the integral of $x^{\frac{5}{4}}$ with respect to $x$ is $\frac{4}{9} x^{\frac{9}{4}}$ .

$\frac{4}{5} (\frac{4}{9} x^{\frac{9}{4}} + C) + \int - \frac{1}{2} x^{2} d x$

Step 4

Since $- \frac{1}{2}$ is constant with respect to $x$ , move $- \frac{1}{2}$ out of the integral.

$\frac{4}{5} (\frac{4}{9} x^{\frac{9}{4}} + C) - \frac{1}{2} \int x^{2} d x$

Step 5

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

$\frac{4}{5} (\frac{4}{9} x^{\frac{9}{4}} + C) - \frac{1}{2} (\frac{1}{3} x^{3} + C)$

Step 6

Simplify.

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Step 6.1

Simplify.

$\frac{4}{5} \cdot \frac{4}{9} x^{\frac{9}{4}} - \frac{1}{2} \cdot \frac{1}{3} x^{3} + C$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{4}{5}$ by $\frac{4}{9}$ .

$\frac{4 \cdot 4}{5 \cdot 9} x^{\frac{9}{4}} - \frac{1}{2} \cdot \frac{1}{3} x^{3} + C$

Step 6.2.2

Multiply $4$ by $4$ .

$\frac{16}{5 \cdot 9} x^{\frac{9}{4}} - \frac{1}{2} \cdot \frac{1}{3} x^{3} + C$

Step 6.2.3

Multiply $5$ by $9$ .

$\frac{16}{45} x^{\frac{9}{4}} - \frac{1}{2} \cdot \frac{1}{3} x^{3} + C$

Step 6.2.4

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

$\frac{16}{45} x^{\frac{9}{4}} - \frac{1}{3 \cdot 2} x^{3} + C$

Step 6.2.5

Multiply $3$ by $2$ .

$\frac{16}{45} x^{\frac{9}{4}} - \frac{1}{6} x^{3} + C$