Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Since $\frac{5}{8}$ is constant with respect to $x$ , move $\frac{5}{8}$ out of the integral.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Reorder terms.

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