Calculus - Old Question 2025 (BT103HS)

1. (a) Evaluate $\int x \log x dx$. [4]

1. (b) Obtain a reduction formula for $\int \sin^{n} x dx$ and find $\int \sin^{6} x dx$. [8]

2. (a) Tangents are drawn from the origin to the curve $y = \sin x$. Prove that their point of contact lie on $x^{2}y^{2} = x^{2} - y^{2}$. [4]

2. (b) If $y = a \cos(\log x) + b \sin(\log x)$, then show that:

• (i) $x^{2}y_{2} + xy_{1} + y = 0$ [4]
• (ii) $x^{2}y_{n+2} + (2n+1)xy_{n+1} + (n^{2}+1)y_{n} = 0$ [4]

3. (a) Evaluate $\iint xy(x+y) dx dy$ over the area between $y = x^{2}$ and $y = x$. [6]

3. (b) Solve the equation $x \cos y dy = (xe^{x} \log x + e^{x}) dx$. [6]

4. Use Gamma function to prove: $$\int_{0}^{\pi} \sin^{6} \frac{x}{2} \cos^{8} \frac{x}{2} dx = \frac{5\pi}{2^{11}}$$ [6]

5. Find the area of the asteroid $x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}$. [6]

6. Find the volume of the solid formed by revolving the cycloid $x = a(\theta + \sin \theta)$, $y = a(1 + \cos \theta)$. [6]

7. Evaluate: $\lim_{x \to 0} \left(\frac{1}{x^{2}} - \frac{1}{\cos^{2} x}\right)$ [6]

8. Find the asymptotes of the curve: $y^{3} - xy^{2} - x^{2}y + x^{3} + x^{2} - y^{2} = 1$. [6]

9. Find the radius of curvature of the curve $r = a(1 - \cos \theta)$. [6]

10. Solve the equation: $$\frac{dy}{dx} + 2y \tan x = \sin x$$ [6]