   Chapter 7.1, Problem 18E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ e − θ cos 2 θ   d θ

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 in terms of u and v is given by

udv=uvvdu

Given:

The integral, eθcos2θdθ.

Calculation:

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

u=cos2θ      dv=eθdθ

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

du=2sin2θdθ      v=eθ

Substitute for variables in the formula above to get

eθcos2θdθ=cos2θ(eθ)(eθ)(2sin2θ)dθ=eθcos2θ2eθsin2θdθ …… (1)

Integration in the last term also involves the product of two functions, so solve it using integration by parts. Then the integration eθsin2θdθ can be done using

u=sin2θ     dv=eθdθ

Then,

du=2cos2θdθ     v=eθ

Substitute the variables into the formula for integration by parts

eθsin2θdθ=sin2θ(eθ)(eθ)(2cos2

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