10th Edition

Ron Larson + 1 other

ISBN: 9781285057095

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.

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)=14∫1tdt=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.

Check out a sample textbook solution.

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

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Calculus Calculus 10th Edition

Finding an Indefinite Integral In Exercises 19-32, find the indefinite integral.

∫ x 3 + x + 1 x 4 + 2 x 2 + 1 d x