18.01 Practice Exam 3

1. (10 points each) Find the following indefinite integrals.

(a) ∫ ( ^{3} \sqrt{x} + 1)^{2} dx.

(b) ∫ \frac{x}{(x^{2} + 1)^{2}} dx.

(c) ∫ (1 + cos^{3} x) sin x dx.

2. (11 points each) Evaluate the following definite integrals.

(a) ∫_{0}^{1}(e^{x} + x^{e}) dx.

(b) ∫_{0}^{π/4} sin(2x) dx.

3. (12 points) If a > 0, then y = a − x^{4} has one arch above the x-axis. Find its area.

4. (12 points) Find the lower sum for ∫_{1}^{2} \frac{1}{x} dx with n intervals.

5. (12 points) Use Simpson’s Rule with n = 4 to approximate the value of

∫_{0}^{2} \frac{1}{4x^{2} + 1} dx.

6. (12 points) Determine the intervals where the function f(x) = ∫_{0}^{x}(e^{t^{2}} − 2) dt is increasing.