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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

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Integration Techniques

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##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

##### Michael J.

Idaho State University

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### Video Transcript

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#### Topics

Integration Techniques

##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

##### Michael J.

Idaho State University

Lectures

Join Bootcamp