Find the integral of − x 2 + 3 x − 2 with respect to x.

Question

Find the integral of   −      x          2            ﻿    +  3  x  −  2 with respect to x.

priffEmipsy4i37m · Accepted Answer

Split the single integral into multiple integrals.

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

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

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

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

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

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

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

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

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

Apply the constant rule.

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

Simplify.

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

Reorder terms.

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

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