   Chapter 7.1, Problem 72E

Chapter
Section
Textbook Problem

# (a) Use integration by parts to show that ∫ f ( x ) d x = x f ( x ) − ∫ x f ′ ( x ) d x (b) If f and g are inverse functions and f′ is continuous, prove that ∫ a b f ( x ) d x = b f ( b ) − a f ( a ) − ∫ f ( a ) f ( b ) g ( y ) d y [Hint: Use part (a) and make the substitution y = f ( x ) . ](c) In the case where f and g are positive functions and b > a > 0 , draw a diagram to give a geometric interpretation of part (b).(d) Use part (b) to evaluate ∫ 1 e ln x   d x .

To determine

(a)

To show: f(x)dx=xf(x)xf(x)dx

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 equation, f(x)dx=xf(x)xf(x)dx

Calculation:

Solve the integration using integration by parts

To determine

(b)

To show: abf(x)dx=bf(b)af(a)f(a)f(b)g(y)dy

To determine

(c)

To draw: the geometric interpretation of result in part (b)

To determine

(d)

To evaluate: 1elnxdx

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