AP EAMCET202125 Aug 2021Evening ShiftMathematicsIndefinite IntegrationActual
If x^2+1 x^4+1 d x=f(x)+c , then f(x) is equal to
Options
- A1 2 ⁻¹ ( x^2+1 2 x )
- B1 2 ⁻¹ ( x^2-1 2 x )
- C1 2 ⁻¹ ( 1-x^2 2 x )
- D1 2 ⁻¹ ( 1+x^4 2 x )
Correct answer
B. 1 2 ⁻¹ ( x^2-1 2 x )
Step-by-step solution
x^2+1 x^4+1 d x=f(x)+c Let aligned & I= x^2+1 x^4+1 d x= x^2 (1+ 1 x^2 ) x^2 (x^2+ 1 x^2 ) d x & I= (1+ 1 x^2 ) (x- 1 x )^2+( 2 )^2 d x aligned Let x- 1 x =t aligned & (1+ 1 x^2 ) d x=d t & I= d t ( 2 )^2+t^2 = 1 2 ⁻¹ ( t 2 )+c & I= 1 2 ⁻¹ ( x^2-1 2 x )+c=f(x)+c & f(x)= 1 2 ⁻¹ ( x^2-1 2 x ) aligned