Why Practice Matters
The Calculus II midterm is a milestone that tests your mastery of integration, infinite series, and applications of calculus to curves and differential equations. While lecture notes give you the theory, solving a variety of problems cements the concepts and reveals hidden pitfalls. This page groups representative practice problems by topic, offers concise solutions, and provides strategic advice for effective study.
Integration Techniques
Problem 1 Integration by Parts
Evaluate the integral xe2x dx.
Solution: Let u = x, dv = e^{2x}dx. Then du = dx, v = (1/2)e^{2x}. Apply the formula udv = uv vdu:
xe^{2x}dx = (x/2)e^{2x} (1/2)e^{2x}dx = (x/2)e^{2x} (1/4)e^{2x} + C.
Problem 2 Trigonometric Substitution
Compute (9 x) dx.
Solution: Set x = 3 sin, dx = 3 cos d. Then (9 x) = 3 cos. The integral becomes 3 cos3 cos d = 9 cos d. Using cos = (1+cos2)/2 gives 9(1+cos2)d = (9/2)( + (1/2)sin2)+C. Convert back: = arcsin(x/3), sin2 = 2 sin cos = 2(x/3)((9x)/3). Finally, (9x)dx = (x/2)(9x) + (9/2)arcsin(x/3) + C.
Problem 3 Partial Fractions
Find (2x+3)/(xx2) dx.
Solution: Factor denominator: (x2)(x+1). Decompose: (2x+3)/(xx2) = A/(x2) + B/(x+1). Solving yields A = 5, B = 3. Integral = 5dx/(x2) 3dx/(x+1) = 5ln|x2| 3ln|x+1| + C.
Series & Sequences
Problem 4 Convergence of a Power Series
Determine the interval of convergence for the series n=1 (x2) / (3 n).
Solution: Use the Ratio Test: L = limn |(x2)3(n/(n+1))| = |x2|/3. Converges if L < 1 |x2| < 3 1 < x < 5. Test endpoints: at x = 1, series becomes (3)/(3 n) = (1)/n, which converges (alternating harmonic). At x = 5, series is 1/n, diverges. Interval of convergence: [1,5).
Problem 5 Taylor Polynomial
Find the fourth-degree Taylor polynomial for f(x)=ln(1+x) centered at 0.
Solution: Derivatives: f'(x)=1/(1+x), f''(x)=1/(1+x), f'''(x)=2/(1+x), f(x)=6/(1+x). Evaluated at 0: 1, 1, 2, 6. Polynomial: P(x)=x x/2 + x/3 x/4 + C (C=0). This matches the known series (1)^{n1} x/n for n=14.
Parametric & Polar Coordinates
Problem 6 Arc Length of a Parametric Curve
For the curve defined by x = t3t, y = t4t on 0 t 1, compute its arc length.
Solution: Arc length = [(dx/dt)+(dy/dt)] dt. dx/dt = 3t3, dy/dt = 4t8t. Square and add: (3t3) + (4t8t) = 9(t2t+1) + 16(t4t+4t) = 16t + (964)t + (18+64)t + 9 = 16t55t+46t+9. The integrand simplifies to ( (4t8t) + (3t3) ). No elementary antiderivative; evaluate numerically: L 3.842 (to three decimal places).
Problem 7 Area in Polar Coordinates
Find the area enclosed by one loop of the rose curve r = 2sin(3).
Solution: A loop occurs when sin(3) 0, i.e., 0 /3. Area = ^{/3} (2sin3) d = 2^{/3} sin3 d. Use sin = (1cos2)/2: = 2^{/3} (1cos6) d = ^{/3} (1cos6) d. Integrate: [ (1/6)sin6]^{/3} = (/3) (1/6)0 = /3. So the area of one loop is /3 square units.
Differential Equations
Problem 8 Solving a FirstOrder Linear ODE
Solve y' + 4y = 8e^{-2x}, given y(0) = 1.
Solution: Integrating factor (x)=e^{4dx}=e^{4x}. Multiply: e^{4x}y' + 4e^{4x}y = 8e^{2x}. Left side is (e^{4x}y)'. Integrate: e^{4x}y = 8e^{2x}dx = 4e^{2x} + C. Hence y = 4e^{-2x} + Ce^{-4x}. Apply y(0)=1 1 = 4 + C C = 3. Final solution: y = 4e^{-2x} 3e^{-4x}.
Problem 9 Separation of Variables
Find the particular solution to dy/dx = ysin x with y(/2) = 1.
Solution: Separate: dy / y = sin x dx. Integrate: -1/y = cos x + C 1/y = cos x + C'. Using point (/2,1): 1 = cos(/2) + C' C' = 1. Thus 1/y = cos x + 1 y = 1/(1 + cos x).
Study Tips for the Midterm
- Active Recall. After reading a theorem, close the book and write it from memory. Then test yourself with a problem that uses that theorem.
- Mix Problem Types. Don't practice only integrals; alternate between series, polar, and differential equations to improve mental flexibility.
- Time Yourself. Simulate exam conditions by solving a set of 57 problems in 45 minutes. Note where you spend too much time and adjust your approach.
- Check the Edge Cases. Many mistakes arise from neglecting endpoints in series or forgetting absolute values in logarithmic integration.
- Summarize Formulas. Create a onepage cheat sheet with integration tricks, common series expansions, and derivative/antiderivative tables. The act of writing the sheet reinforces memory.
- Teach a Peer. Explaining a solution aloud forces you to clarify each step, exposing gaps you might miss while working silently.
