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(a) Use integration by parts to show that$$ \int f(x) dx = xf (x) - \int xf^\prime (x) dx $$(b) If $ f $ and $ g $ are inverse functions and $ f^\prime $ is continuous, prove that$$ \int_a^b f(x) dx = bf (b) - af (a) - \int_{f(a)}^{f(b)} g(y) dy $$[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 $ \displaystyle \int_1^e \ln x dx $.

a) Choose $u=f(x), d v=d x,$ find $d u$ and $v,$ then put everything into the integration by parts formula.b) $\int_{a}^{b} f(x) d x=b f(b)-a f(a)-\int_{f(a)}^{f(b)} g(y) d y$c) See the imaged) 1

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 1

Integration by Parts

Integration Techniques

Oregon State University

Harvey Mudd College

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Boston College

Lectures

01:11

In mathematics, integratio…

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In grammar, determiners ar…

03:03

A useful integrala. Us…

02:21

Prove that $\int_{a}^{b} \…

03:28

(a) If $ f $ is continuous…

06:26

Product of integrals Suppo…

02:29

Integrating derivatives Us…

00:49

If $ f $ is continuous on …

05:08

Show that if $f$ and $g$ a…

02:52

Rewriting Integrals Assume…

00:42

06:28

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