   Chapter 7.1, Problem 30E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 1 3 arctan ( 1 / x )   d x

To determine

To evaluate: The given integral using the technique of integration by parts.

Explanation

The technique of integration by parts comes in handy when the integrand involves product of two functions. It can be thought of as a rule corresponding to the product rule in differentiation.

Formula used:

The formula for integration by parts for definite integral is given by

abf(x)g(x)dx=f(x)g(x)]ababg(x)f(x)dx

Given:

The integral, 13arctan(1/x)dx.

Calculation:

Make the choice for f=u and g’=dv such that the resulting integration from the formula above is easier to integrate. Let

u=arctan(1x)      dv=dx

Then, the differentiation of u and antiderivative of dv will be

du=11+(1x)2     v=x

Using the formula above, the given integration will become

13arctan(1/x)dx=arctan(1x)x]1313x11+(1x)2dx

Let x2=t,2xdx=dt, then the limits will be

x1,t1x3,t3

The above integral will take the form

13arctan(1/x)dx=arctan(1t)t]13121311+1tdt=arctan(1t)t]131213tt+1dt …… (1)

The integral in last term obtained above can be solved with integration by parts

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