MTH 252 Winter 2024 Written Homework 4 Due March 1 Name: You are encouraged to work on this assignment collaboratively with your peers in class. You can also ask me for hints if you are stuck. Yet you should write up your own work and submit it on Canvas in ONE pdf file. Show all of your work for full credit, and your work should be clearly written and organized. The homework is about 7.1, 7.3, and 7.5. 1 : Problem 1. Evaluate the definite integral / e dx 0 Hint: We can rewrite 2 into 2 - , and what is / xe " da? Problem 2. Evaluate the indefinite integral f cos (V1) dx. Hint: This integral is harder than it seems. You will need BOTH the u-sub and the in- tegration by parts. So which method should we use first on this COMPOSITE function? Also, remember the trick you used on Problem 2(b) in Written Homework 3? That will be useful! Problem 3. A lot of you used a formula in written homework 3 that we haven't justified yet so let's prove that formula! (a) Using a trigonometric substitution, compute the integral and show that for a > 0, dx 1 L /T __ = - ts I (j 1 / x? + a? 4} & (a) + 1) (b) Verify the formula above via differentiation. - 22 oy Problem 4. A first-year Calculus student is trying to evaluate the integral / % as fol- Ak T [ lows: Since the denominator is factored out, we can set up a partial fraction decomposition to break up the fraction. z? A B (z 4)(z + 5) :r:4+:{r+5 Do you think his approach will work out? If so, what is the constant A and B? If not, what goes wrong with it? Justify your reasoning. Hint: Let's try to follow the procedure of the method of partial fraction decomposition. Can you find a value for A and a value of B