   Chapter 7.1, Problem 34E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 0 1 r 3 4 + r 2   d r

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, 01r34+r2dr.

Calculation:

Rewrite the above integration using the substitution 4+r2=t. Then,

2rdr=dt and r2=t4

The limits of integration will change as

r0,t4r1,t5

The integration in terms of t will be

01r34+r2dr=1201r24+r22rdr=1245t4tdt …… (1)

Now, solve the integration 45t4tdt. Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate

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