If ∫₀^(π/3) cos ⁴ x d x=a π+b √3, where a and b are rational numbers, then 9 a+8 b is equal to :

Solution

Maths sir · 4 steps

Cut cos⁴ x down to first powers of cosines with the double-angle formula, then integrate term by term.

1. Reduce the power. Use cos² x = (1 + cos 2x)/2 twice. Squaring gives (1 + 2 cos 2x + cos² 2x)/4, and then cos² 2x = (1 + cos 4x)/2.

   cos⁴ x = (3 + 4 cos 2x + cos 4x)/8

2. Integrate. The antiderivative is 1/8[3x + 2 sin 2x + (sin 4x)/4], and it is 0 at x = 0.
3. Put in the limit. At x = π/3: 3x = π, 2 sin 2π/3 = √3 and 1/4 sin 4π/3 = -√3/8. So the integral is 1/8(π + 7√3/8) = π/8 + 7√3/64.
4. So. a = 1/8 and b = 7/64, so 9a + 8b = 9/8 + 7/8 = 2, which is option (a).

The idea to keep. For even powers of sine or cosine, keep using cos² θ = (1+cos 2θ)/2 until every term has power one. Watch the sign of sin past π.

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