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Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.

x3-3x2+2x-3x2+1dx

Short Answer

Expert verified

x223x+12lnx2+1+C.

Step by step solution

01

Step1. Given Information

The integral is as follows.

x33x2+2x3x2+1dx

The objective is ton solve the integral.

02

Step2. Long division

The polynomial long division method is calculated below.

x-3x2+1x33x2+2x3x3+x3x2+x3x23x

The expression is in the form of x33x2+2x3x2+1=x3+xx2+1

03

Step3. Solution

The integral is solved below.

x33x2+2x3x2+1dx=x3+xx2+1dx=xdx3dx+xx2+1dx=x223x+121uduu=x2+1,du=2xdx=x223x+12ln(|u|)+Cx223x+12lnx2+1+C

Therefore, the value is x223x+12lnx2+1+C.

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