Exam Questions & Step-by-Step Markschemes (11 Problems)
Question 1 • 2024
4 Marks
Evaluate:\n\n \int e^{2x} \sin(3x) \, dx
Answer: I = \frac{e^{2x}}{13} (2\sin 3x - 3\cos 3x) + C
Question 2 • 2024
4 Marks
Evaluate: I = \int_{0}^{\pi/2} \ln(\sin x) \, dx
Answer: I = -\frac{\pi}{2}\ln 2
Question 3 • 2024
4 Marks
Evaluate the definite integral: \int_{2}^{8} \frac{\sqrt{10 - x}}{\sqrt{x} + \sqrt{10 - x}} \, dx
Answer: I = 3
Question 4 • 2024
4 Marks
Evaluate the definite integral: \int_{0}^{\pi} \frac{x \sin x}{1 + \cos^2 x} \, dx
Answer: I = \frac{\pi^2}{4}
Question 5 • 2024
4 Marks
Evaluate the definite integral: I = \int_{0}^{\frac{\pi}{4}} \log(1 + \tan x) \, dx
Answer: I = \frac{\pi}{8} \log 2
Question 6 • 2024
4 Marks
Find the indefinite integral using the method of partial fractions: \int \frac{x}{(x - 1)(x^2 + 1)} \, dx
Answer: \frac{1}{2}\ln|x - 1| - \frac{1}{4}\ln(x^2 + 1) + \frac{1}{2}\tan^{-1} x + C