If the integral 525 ∫0^(π/2) sin 2 x cos ^(11/2) x(1+Cos^(5/2) x)^(1/2) d x is equal to (n √2-64), then n is equal to _________.

Solution

Maths sir · 4 steps

Split sin 2x into 2 sin x cos x, put t=cos x, then let u = 1 + t^(5/2) to clear the root.

1. First substitution. Write sin 2x = 2 sin x cos x, so the integrand becomes 2 sin x cos^(13/2) x (1+cos^(5/2) x)^(1/2). Put t=cos x, so dt = -sin x dx and the limits 0 → π/2 become 1 → 0. The minus sign flips them back, so I = 1050∫₀¹ t^(13/2)(1+t^(5/2))^(1/2) dt.
2. Second substitution. Let u = 1 + t^(5/2), so du = 5/2t^(3/2) dt, that is t^(3/2) dt = 2/5 du. Split t^(13/2) = t⁵ · t^(3/2) and use t⁵ = (u-1)². The limits become 1 → 2.

   I = 1050 × 2/5∫₁² (u-1)² u^(1/2) du = 420[2/7u^(7/2) - 4/5u^(5/2) + 2/3u^(3/2)]₁²

3. Evaluate. At u=2: (16/7 - 16/5 + 4/3)√2 = 44/105√2. At u=1: 2/7 - 4/5 + 2/3 = 16/105. So I = 420 × (44√2 - 16)/105 = 4(44√2 - 16).
4. So. I = 176√2 - 64. Matching this with n√2 - 64 gives n = 176.

The common slip: 88. Dropping the factor 2 from sin 2x = 2 sin x cos x halves everything and gives 88, not 176.

The idea to keep. When a power sits under a root, substitute the whole bracket. Then split the leftover power so that one piece is exactly the derivative you need for du.

Answer from the NTA final answer key. Solution by PrepWiser.