The curve C has polar equation r = a(1 + sin θ), where a is a positive constant and 0 ≤ θ < 2π. Draw a sketch of C. Find the exact value of the area of the region enclosed by C and the half-lines θ = π/4 and θ = π.

The curves C₁ and C₂ have polar equations C₁: r = 1/√2, for 0 < θ < 2π, C₂: r = √(sin10), for 0 ≤ θ≤ π. Find the pola…

The curve C has polar equation r = a(1 + sin θ) for –π < θ < π, where a is a positive constant.

The curve C has polar equation r = cos 2θ, for -¼π ≤ θ ≤ ¼π.

It is given that S_n = ∑_{r=1}^n u_r = 2n² + n. Write down the values of S1, S2, S3, S4. Express u_r in terms of r, j…

It is given that I_n = ∫_0^1 x^n / √(1 + 2x) dx. Show that, for n ≥ 1, (2n + 1)I_n = √3 - nI_{n-1}. Show that I_3 = 2…

It is given that y = (1 + x)² ln(1 + x). Find d³y/dx³. Prove by mathematical induction that, for every integer n ≥ 3,…