Exam Questions & Step-by-Step Markschemes (249 Problems)
Question 1 • 2022
2 Marks
If \frac{d}{dx}[F(x)] = \frac{\sec^4 x}{\mathrm{cosec}^4 x} and F\left(\frac{\pi}{4}\right) = \frac{\pi}{4} , then find F(x) .
Answer: F(x) = \frac{\tan^3 x}{3} - \tan x + x + \frac{2}{3}
Question 2 • 2022
2 Marks
Find : \int \frac{\log x - 3}{(\log x)^4}\,dx .
Answer: \frac{x}{(\log x)^3} + C
Question 3 • 2022
3 Marks
Find : \int \frac{dx}{\sqrt{x} + \sqrt[3]{x}} .
Answer: 2\sqrt{x} - 3\sqrt[3]{x} + 6\sqrt[6]{x} - 6\log\left(1 + \sqrt[6]{x}\right) + C
Question 4 • 2022
3 Marks
Evaluate : \int_0^{\pi/2} \frac{\cos x}{(1 + \sin x)(4 + \sin x)}\,dx .
Answer: \frac{1}{3}\log\frac{8}{5}
Question 5 • 2022
4 Marks
Evaluate : \int_0^{\pi} \frac{x}{1 + \sin x}\,dx .
Answer: \pi
Question 6 • 2022
3 Marks
Find : \int \frac{x^3 + x}{x^4 - 9}\,dx .
Answer: \frac{1}{4}\log|x^4 - 9| + \frac{1}{12}\log\left|\frac{x^2 - 3}{x^2 + 3}\right| + C