Question 306.
Suppose that ∫(Π/4 to Π/2) cos^4x/sin^5x dx= k.
Find the value ∫(Π/4 to Π/2) cos^6x/sin^7x dx and
express your answer in terms of k.
Simplify your final answer as much as possible.
Answer 306.
∫ cos^6 x / sin^7 x dx
= ∫ cos^5 x . (sin^-7 x cosx) dx
= cos^5 x ∫ sin^-7 x d(sinx) - ∫ [d/dx(cos^5 x)
. ∫ sin^-7 x d(sinx)] dx
= - (1/6) cos^5 x sin^-6 x - ∫ (- 5cos^4 x sinx) . (- sin^-6 x / 6) dx
= - (1/6) cos^5 x sin^-6 x - (5/6) ∫ (cos^4 x / sin^5 x) dx
Plugging limits x = π/4 to π/2,
∫ cos^6 x / sin^7 x dx ... (x = π/4 to π/2)
= - (1/6) cos^5 x sin^-6 x ... (x = π/4 to π/2) - 5k/6
= - (1/6) [0 - (1/√2)^5 * (1/√2)^-6] - 5k/6
= (1/6) (√2 - 5k).
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Suppose that ∫(Π/4 to Π/2) cos^4x/sin^5x dx= k.
Find the value ∫(Π/4 to Π/2) cos^6x/sin^7x dx and
express your answer in terms of k.
Simplify your final answer as much as possible.
Answer 306.
∫ cos^6 x / sin^7 x dx
= ∫ cos^5 x . (sin^-7 x cosx) dx
= cos^5 x ∫ sin^-7 x d(sinx) - ∫ [d/dx(cos^5 x)
. ∫ sin^-7 x d(sinx)] dx
= - (1/6) cos^5 x sin^-6 x - ∫ (- 5cos^4 x sinx) . (- sin^-6 x / 6) dx
= - (1/6) cos^5 x sin^-6 x - (5/6) ∫ (cos^4 x / sin^5 x) dx
Plugging limits x = π/4 to π/2,
∫ cos^6 x / sin^7 x dx ... (x = π/4 to π/2)
= - (1/6) cos^5 x sin^-6 x ... (x = π/4 to π/2) - 5k/6
= - (1/6) [0 - (1/√2)^5 * (1/√2)^-6] - 5k/6
= (1/6) (√2 - 5k).
Link to YA!
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