   Chapter 8.4, Problem 30E

Chapter
Section
Textbook Problem

Finding an Indefinite Integral In Exercises 19-32, find the indefinite integral. ∫ x 3 + x + 1 x 4 + 2 x 2 + 1   d x

To determine

To calculate: The solution of the indefinite integral x3+x+1x4+2x2+1dx.

Explanation

Given:

The indefinite integral is x3+x+1x4+2x2+1dx

Formula used:

Integration identity, 1f(x)(df(x))=lnf(x)+c

Calculation:

The indefinite integral given in the question can be rewritten as;

x3+x+1x4+2x2+1dx=x3+x+1(x2+1)2dx=x3+x(x2+1)2dx+1(x2+1)2dx=x3+xx4+2x2+1dx+1(x2+1)2dx

Here, consider x4+2x2+1=t so, the derivative of the equation is;

(4x3+4x)dx=dt4(x3+x)dx=dt(x3+x)dx=dt4dx=dt4(x3+x)

So, the integration of first part x3+xx4+2x2+1dx is;

x3+xx4+2x2+1dx=x3+xtdt4(x3+x)=141tdt=14lnt+C1=14ln|(x4+2x2+1)|+C1

Now, for the integration of second part 1(x2+1)2dx. So, make use of the trigonometric solution, x=tanθ to solve it as shown in figure.

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