Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

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

Step 4

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

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

Step 5

Since $\frac{2}{3}$ is constant with respect to $x$ , move $\frac{2}{3}$ out of the integral.

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

Step 6

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

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

Step 7

Apply the constant rule.

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Reorder terms.

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