18.01 PROBLEM SET 5

Hand in to Room 2-108 by 1:00 pm Wednesday, April 10.

The due date and time are absolute. Late homeworks are accepted until the solutions are posted, but the score will be multiplied by 1/2.

Everything included here may be used later. You do not need to memorize the proofs, but you need to know the facts.

1. FT.8/3, 7, 10. 282/40, 45, 49. 286/7, 15, 21. 290/10, 25, 29.

(15 points)

2. Explain with your own words the geometric meaning of changing variables in

definite integrals. Why do we have to change the limits of integration? (2 points)

3.

(a) If y = ln x, then x = e^{y}. Show that

∫_{1}^{b} ln xdx + ∫_{0}^{ln b} e^{y}dy = b ln b

by representing the two definite integrals as areas on the graph of y = ln x.

(b) Find a formula for

∫_{1}^{b} ln xdx

by evaluating the second integral in (a).

(c) Prove the formula in (b) directly without using geometry by differentiating with respect to b. To complete the proof you need to show more than just that the derivatives are equal.

(d) For any continuous function f, find the formula for

\frac{d}{db} ∫_{0}^{ln b} f(y)dy

and confirm it in the special case f(y) = e^{y}, which you evaluated directly earlier in this problem.

(8 points)

4. Use the x = tan θ substitution to compute ∫ \frac{dx}{\sqrt{1 + x^{2}}} . In the second problem

set we computed \frac{d}{dx} sinh^{−1} x. Combining these two results obtain an explicit

expression for sinh^{−1} x in terms of x.

(5 points)

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