∫0^(π / 4) (cos ² x sin ² x)/(cos ³ x+sin ³ x)² d x is equal to

Solution

Maths sir · 3 steps

Divide top and bottom by cos⁶ x, and the whole thing becomes a function of tan³ x.

1. Divide by cos^6. The denominator has degree 6 in sine and cosine, so divide numerator and denominator by cos⁶ x. The numerator becomes (sin² x)/(cos² x) × 1/(cos² x) = tan² x sec² x, and the denominator becomes (1 + tan³ x)².
2. Substitute. Put t = tan³ x, so dt = 3 tan² x sec² x dx. The limits change from x = 0 to t = 0 and from x = π/4 to t = 1.

   ∫₀^(π/4) (tan² x sec² x)/(1+tan³ x)² dx = 1/3∫₀¹ dt/(1+t)² = 1/3[-1/(1+t)]₀¹

3. So. The bracket is -1/2 + 1 = 1/2, so the integral is 1/3 × 1/2 = 1/6. The answer is (b).

The idea to keep. When every term in a trigonometric integrand has the same total degree in sine and cosine, divide by the matching power of cos x. That turns it into tan x and sec² x, ready for a substitution.

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