Question

int_x cos^4x/ int_0 cos^4x + sin^4x dx =

1) pi/4
2) pi/2
3) pi/8
4) pi

Answer 1

Step-by-step explanation:

we will use following properties of definite integral to solve this problem. As it is difficult to use integration mark, I'll write the word int for integration

property

int (limit 0 to 2a) f(x)dx = 2*int(limit 0 to a) f(x)DX

(if f(2a-x) = f(x))

first we will verify if the condition is true in this case

the limit given is from 0 to π

so 2a = π, a= π/2

sin^4(π-x) = sin^4x

cos^4(π-x) = (-cosx)^4 = cos^4x

hence we can see that the given function will comply with the condition of f(2a-x) = f(x)

now we change the integration as

I = 2*int(limit 0 to π/2) cos^4xdx/(cos4x+sin4x).....(1)

now we will use the propert

int(limit. 0 to a)f(x)dx = int (limit 0 to a) f(a-x)dx

thus

I = 2*int(lim 0. to π/2). cos^4(π/2-x)dx/(cos^4(π/2-x)+sin^4(π/2-x)

= 2*int(limit 0 to π/2) sin^4xdx/(sin^4x+cos^4x)...(2)

adding eqn (1) and (2)

2I = 2*int(limit 0 to π/2) (cos^4x+sin^4x)dx/(cos^4x+sin^4x)

2I = 2*int(limit 0 to π/2) dx

I = (x) (0 to π/2)

I = π/2

hope you find it useful.