Find the integral of x 5 − x 3 + 3 with respect to x.

Question

Find the integral of       x          5            ﻿    −      x          3        +  3 with respect to x.

Duncan Cox · Accepted Answer

Split the single integral into multiple integrals.

$\int x^{5} d x + \int - x^{3} d x + \int 3 d x$

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$\frac{1}{6} x^{6} + C + \int - x^{3} d x + \int 3 d x$

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{6} x^{6} + C - \int x^{3} d x + \int 3 d x$

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

$\frac{1}{6} x^{6} + C - (\frac{1}{4} x^{4} + C) + \int 3 d x$

Apply the constant rule.

$\frac{1}{6} x^{6} + C - (\frac{1}{4} x^{4} + C) + 3 x + C$

Simplify.

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

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