AP EAMCET202423 May 2024Morning ShiftMathematicsIndefinite IntegrationActual
If A= ₀^ 1+x^2 1+x^4 d x, B= ₀^1 1+x^2 1+x^4 d x , then
Options
- A2 ~A = B
- BA = B
- C2 ~B = A
- D2 ~B + A =0
Correct answer
C. 2 ~B = A
Step-by-step solution
A= ₀^ 1+x^2 1+x^4 d x, B= ₀^1 1+x^2 1+x^4 d x Let I= 1+x^2 1+x^4 d x= 1+x^2 (x^2- 2 x+1 ) (x^2+ 2 x+1 ) d x By partial fraction aligned & I= ( 1 2 (x^2+ 2 x+1 ) + 1 2 (x^2- 2 x+1 ) ) d x & I= 1 2 1 x^2+ 2 x+1 d x+ 1 2 1 x^2- 2 x+1 d x & I= 1 2 d x (x+ 1 2 )^2+ 1 2 + 1 2 d x (x- 1 2 )^2+ 1 2 & I= 1 2 2 ⁻¹ (x+ 1 2 ) 1 2 & + 1 2 2 ⁻¹ (x- 1 2 ) 1 2 +C aligned aligned & I= 1 2 ⁻¹( 2 x+1)+ 1 2 ⁻¹( 2 x-1)+C & I= 1 2 ⁻¹ 2 x 1-x^2 +C & A=[I]₀^ = 1 2 [ ⁻¹ 2 x 1-x^2 ]₀^ = 2 & B=[I]₀^1= 1 2 [ ⁻¹ 2 x 1-x^2 ]₀^1= 2 2 & 2 B=A ali