Calculus Examples

$\int x^{- 3} - 3 x^{\frac{1}{4}} + 8 x^{2} d x$

Step 1

Split the single integral into multiple integrals.

$\int x^{- 3} d x + \int - 3 x^{\frac{1}{4}} d x + \int 8 x^{2} d x$

Step 2

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

$- \frac{1}{2} x^{- 2} + C + \int - 3 x^{\frac{1}{4}} d x + \int 8 x^{2} d x$

Step 3

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

$- \frac{1}{2} x^{- 2} + C - 3 \int x^{\frac{1}{4}} d x + \int 8 x^{2} d x$

Step 4

By the Power Rule, the integral of $x^{\frac{1}{4}}$ with respect to $x$ is $\frac{4}{5} x^{\frac{5}{4}}$ .

$- \frac{1}{2} x^{- 2} + C - 3 (\frac{4}{5} x^{\frac{5}{4}} + C) + \int 8 x^{2} d x$

Step 5

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$- \frac{1}{2} x^{- 2} + C - 3 (\frac{4}{5} x^{\frac{5}{4}} + C) + 8 \int x^{2} d x$

Step 6

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

$- \frac{1}{2} x^{- 2} + C - 3 (\frac{4}{5} x^{\frac{5}{4}} + C) + 8 (\frac{1}{3} x^{3} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$- \frac{1}{2 x^{2}} - \frac{12 x^{\frac{5}{4}}}{5} + 8 (\frac{1}{3} x^{3}) + C$

Step 7.2

Simplify.

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Step 7.2.1

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

$- \frac{1}{2 x^{2}} - \frac{12 x^{\frac{5}{4}}}{5} + 8 \frac{x^{3}}{3} + C$

Step 7.2.2

Combine $8$ and $\frac{x^{3}}{3}$ .

$- \frac{1}{2 x^{2}} - \frac{12 x^{\frac{5}{4}}}{5} + \frac{8 x^{3}}{3} + C$

Step 8

Reorder terms.

$- \frac{1}{2 x^{2}} - \frac{12}{5} x^{\frac{5}{4}} + \frac{8}{3} x^{3} + C$