Q.248. Some simple integrations.

Question 248.
Find the integral with respect to x of
1) 1 / [cos(x-a)*cos(x-b)]
2) 1 / [sin(x)*cos^3(x)]
3) cos(2x) - cos(2A) / [cos(x) - cos(A)].

Answer 248.
1)
∫ dx / [cos(x-a)*cos(x-b)]
= 1/sin(b-a) ∫ sin[(x-a) - (x - b)] dx / [cos(x-a)*cos(x-b)]
= 1/sin(b-a) ∫ [sin(x-a) cos(x-b) - cos(x-a)sin(x-b)] dx / [cos(x-a)*cos(x-b)]
= 1/sin(b-a) ∫ [tan(x-a) - tan(x-b)] dx
= cosec(b-a) * ln [sec(x-a) / (sec(x-b)] + c.

2)
∫ dx / [sinx * cos^3 x]
= ∫ 2sec^2 x dx / [2sinx cosx]
= ∫ 2sec^2 x dx / sin2x
= ∫ 2sec^2 x dx / [2tanx/(1 + tan^2 x)]
= ∫ (1 + tan^2 x) sec^2 x dx / tanx

Let tanx = t => sec^2 x dx = dt
=> Integral
= ∫ (1 + t^2) / t dt
= ∫ (1/t + t) dt
= ln l t l + t^2/2 + c
= ln l tanx l + (1/2) tan^2 x + c.

3)
(cos2x - cos2A) / (cosx - cosA)
= (2cos^2 x - 1 - 2cos^2 A + 1) / (cosx - cosA)
= 2 (cos^2 x - cos^2 A) / (cosx - cosA)
= 2 [(cosx - cosA) * (cosx + cosA)] / (cosx - cosA)
= 2 (cosx + cosA)
=> Integral
= 2 ∫ (cosx + cosA) dx
= 2(sinx + x cosA) + c.

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