18.01 Practice Exam 5

1. (5 points each) Evaluate:

(a) tan^{−1} 1;

(b) cosh(ln 2).

2. (5 points each)

(a) For what values of of x is f(x) = sin^{−1}(e^{x}) defined?

(b) Find f′(x).

3. (10 points each) Evaluate the following definite integrals:

(a) ∫_{0}^{π/4} cos^{5} x dx;

(b) ∫_{1}^{e} \sqrt{x} ln x dx.

4. (10 points each) Find the following indefinite integrals: (a) ∫ x cosh(x^{2}) dx;

(b) ∫ \frac{1}{\sqrt{2x − x^{2}}} dx;

(c) ∫ \frac{1}{x^{3} − x^{2}} dx;

(d) ∫ \frac{x^{2}}{\sqrt{a^{2} − x^{2}}} dx;

5. (10 points) Use the reduction formula

∫ sec^{n} u du = \frac{1}{n − 1} sec^{n−2} u tan u + \frac{n − 2}{n − 1} ∫ sec^{n−2} u du

to find ∫ \frac{x^{4}}{\sqrt{x^{2} − 1}} dx.

6. (5 points each)

(a) State the formula for integration by parts (assume u and v are differentiable functions of x).

(b) Derive the formula from a basic differentiation rule.