Calculus I, Fall 2013 (http://www.math.nthu.edu.tw/˜ wangwc/)

Brief answer to selected problems in Homework 15

1. Section 8.2

Problem 28: With the hint, = ∫0^{\frac{1}{2}} \frac{ds}{\sqrt{1 − s}} = 2 − \sqrt{2}.

Problem 34: Sol 1: ∫ sec x tan^{2} xdx = ∫ tan xd sec x = tan x sec x − ∫ sec^{3} xdx

= tan x sec x − ∫ sec x tan^{2} xdx − ∫ sec xdx.

Sol 2: ∫ sec x tan^{2} xdx = ∫ \frac{sin^{2} x}{cos^{3} x}dx = ∫ \frac{s^{2}ds}{(1 − s^{2})^{2}} . Then apply technique of partial fractions in Section 8.4.

2. Section 8.3 Problem 48: Using u^{2} = x − 1 and Trigonometric Substitutions. Problem 54: Trigonometric Substitutions can get answer: abπ