AP EAMCET202318 May 2023Morning ShiftMathematicsIndefinite IntegrationActual
If x (1-x^3 )^ -1 2 d x= 2 3 g(f(x))+c , then
Options
- Af(x)= x , g(x)= ⁻¹ x
- Bf(x)=x^ 3 2 , g(x)= ⁻¹ x
- Cf(x)=x^ 3 2 , g(x)= ⁻¹ x
- Df(x)= x , g(x)= ⁻¹ x
Correct answer
B. f(x)=x^ 3 2 , g(x)= ⁻¹ x
Step-by-step solution
I= x (1-x^3 )^ - 1 2 d x Let x ^ 3 2 = t 3 2 x ^ 1 2 dx = dt x dx = 2 3 dt I = (1- t ^2 )^ - 1 2 2 3 dt = 2 3 1 1- t ^2 dt aligned & I= 2 3 ⁻¹ t+c & I= 2 3 ⁻¹ (x^ 3 2 )+c aligned Let f(x)=x^ 3 / 2 & g(x)= ⁻¹ x Then, I= 2 3 g(f(x))+c