If ∫π30cos4xdx=aπ+b3√, where a and b are rational numbers, then 9a+8b is equal to :

To solve this problem, we need to use the power-reduction identity for cosine, which is cos^2(x) = (1 + cos(2x))/2.

Step 1: Rewrite cos^4(x) as (cos^2(x))^2.

Step 2: Substitute the power-reduction identity into the integral: ∫π/3 0 (1 + cos(2x))^2 dx.

Step 3: Expand the square: ∫π/3 0 (1 + 2cos(2x) + cos^2(2x)) dx.

Step 4: Use the power-reduction identity again to rewrite cos^2(2x) as (1 + cos(4x))/2.

Step 5: Substitute this back into the integral: ∫π/3 0 (1 + 2cos(2x) + 1/2 + 1/2 cos(4x)) dx.

Step 6: Simplify the integral: ∫π/3 0 (3/2 + 2cos(2x) + 1/2 cos(4x)) dx.

Step 7: Now, we can integrate term by term: [3/2x + sin(2x) + 1/8 sin(4x)] evaluated from 0 to π/3.

Step 8: Substitute the limits of integration: 3/2 * π/3 + sin(2π/3) + 1/8 sin(4π/3) - (3/2 * 0 + sin(0) + 1/8 sin(0)).

Step 9: Simplify the expression: π/2 + √3/2 - 0 = π/2 + √3/2.

So, a = 1/2 and b = 1/2.

Finally, substitute a and b into the expression 9a + 8b to get: 9/2 + 4 = 13/2 = 6.5.