Let f(a) = _ - ^ dx (1+5^x)(a ^2 x + ^2 x)^2 for a > 0 . If f'(3) = - k 3 , then the value of k is

Step-by-step solution

f(a) = _ - ^ dx (1+5^x)(a ^2 x + ^2 x)^2 Using the property _ -c ^ c g(x) dx = ₀^ c (g(x) + g(-x)) dx , we get: f(a) = ₀^ dx (a ^2 x + ^2 x)^2 Since the integrand is symmetric about 2 , we use ₀^ 2c g(x) dx = 2 ₀^ c g(x) dx : f(a) = 2 ₀^ 2 dx (a ^2 x + ^2 x)^2 Dividing numerator and denominator by ^4 x : f(a) = 2 ₀^ 2 ^4 x , dx (a ^2 x + 1)^2 = 2 ₀^ 2 (1+ ^2 x) ^2 x , dx (a ^2 x + 1)^2 Let x = t ^2 x , dx = dt . f(a) = 2 ₀^ 1+t^2 (at^2+1)^2 dt Substitute t = 1 a dt = 1 a ^2 , d . Limits change from 0 to 0 2 . f(a)