Chapter 8.4, Problem 30E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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 âˆ«x3+x+1x4+2x2+1dx

Formula used:

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

Calculation:

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

âˆ«x3+x+1x4+2x2+1dx=âˆ«x3+x+1(x2+1)2dx=âˆ«x3+x(x2+1)2dx+âˆ«1(x2+1)2dx=âˆ«x3+xx4+2x2+1dx+âˆ«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 âˆ«x3+xx4+2x2+1dx is;

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

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

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