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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
Baylor University
Boston College
Lectures
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