Module 8 Quiz: Techniques of Integration
Quiz Instructions
Answer each question, then reveal the answer. Aim for at least 7 out of 10 correct.
1. Evaluate ∫ (2x + 3)7 dx.
u = 2x + 3, du = 2 dx. (1/2) · u8/8 + C = (2x + 3)8/16 + C.
2. Evaluate ∫0π/2 sin x cos²x dx.
u = cos x, du = −sin x dx. −∫10 u² du = ∫01 u² du = 1/3. Answer: 1/3.
3. Evaluate ∫ cos²x dx.
= ∫ (1 + cos 2x)/2 dx = x/2 + sin(2x)/4 + C.
4. Using IBP, evaluate ∫ x ln x dx.
u = ln x, dv = x dx. = (x²/2) ln x − ∫ x/2 dx = (x²/2) ln x − x²/4 + C = (x²/4)(2 ln x − 1) + C.
5. Evaluate ∫ x sin(2x) dx.
u = x, dv = sin 2x dx. v = −cos(2x)/2. = −x cos(2x)/2 + (1/2)∫ cos(2x) dx = −x cos(2x)/2 + sin(2x)/4 + C.
6. What is the LIATE rule and why is it useful?
LIATE stands for Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. It is a guideline for choosing u in integration by parts: pick the function type that appears earliest in LIATE as u, because those functions simplify when differentiated.
7. Evaluate ∫02 xe−x dx.
IBP: u = x, dv = e−x dx. [−xe−x − e−x]02 = (−2e−2 − e−2) − (0 − 1) = 1 − 3e−2 ≈ 0.594.
8. What pattern of coefficients does Simpson's Rule use?
1, 4, 2, 4, 2, ..., 4, 1 (with factor Δx/3 in front). n must be even. It approximates the integrand with parabolas and has error O(1/n4).
9. Evaluate ∫ e√x / √x dx.
u = √x, du = 1/(2√x) dx. So dx/√x = 2 du. 2 ∫ eu du = 2e√x + C.
10. True or false: Simpson's Rule gives the exact answer for any cubic polynomial.
True. Simpson's Rule is exact for polynomials of degree 3 or less because the fourth derivative (which appears in the error bound) is zero for such polynomials.