Problems

Evaluate the integrals in Problems 1-32.

In Problems 33-3θ, express the given integral in terms of

∫ tan x dx,         ∫ cot x dx,         ∫ sec x dx,         ∫ csc x dx.

40           Check the reduction formula for ∫ sinⁿ x dx by differentiating both sides of the equation. Do the same for ∫ tanⁿ x dx and ∫ secⁿ x dx.

41            Find a reduction formula for ∫ xⁿ sin x dx using integration by parts.

42             Find the volume of the solid generated by rotating the region under the curve y = sin²x, 0 ≤ x ≤ π, about (a) the x-axis, (b) the y-axis.

43            Find the volume of the solid generated by rotating the region under the curve y = sin x cos x, 0 ≤ x ≤ π/2, about (a) the x-axis, (b) the y-axis.