Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 4

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

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

Step 5

The integral of $x^{- 1}$ with respect to $x$ is $\ln (| x |)$ .

$8 (\frac{1}{4} x^{4} + C) - 2 (\ln (| x |) + C) + \int \sqrt{x} d x$

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{4}$ and $x^{4}$ .

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

Step 8.2

Simplify.

$2 x^{4} - 2 \ln (| x |) + \frac{2}{3} x^{\frac{3}{2}} + C$