Integration

Find (a) ∫ tan² x dx (b) ∫₀^π/4 sec² xtan³ x dx (c) ∫ 1/(xln x) dx (d) ∫₁² 1/((2x + 1)²) dx

(a) The curves y = sin x and y = cos x intersect at x = π/4 and x = 5π/4. Find the exact area enclosed between the curves for π/4 ≤ x ≤ 5π/4. (b) Use integration by parts to find the exact value of ∫₀^π/4 xcos 2x dx.

The height, h metres, of a tree t years after planting is modelled by dh/dt = (8 - h)/10. When planted, the tree is 1 m tall. (a) Solve the differential equation to find h in terms of t. (b) Find the time taken for the tree to reach 6 m. (c) State the height the model predicts the tree will approach, and give one criticism of the model.

(a) Use the substitution x = 2sin θ to find the exact value of ∫₀¹ 1/(√(4 - x²)) dx. (b) Use the substitution u = √x to find the exact value of ∫₀⁴ e^√x dx.