AP EAMCET202017 Sep 2020Morning ShiftMathematicsIndefinite IntegrationActual
( x^3-1 x^3+x d x= )
Options
- A(x+ |x|+ 1 2 (x^2+1 )+ ⁻¹(x)+c )
- B(x- |x|+ 1 2 (x^2+1 )- ⁻¹(x)+c )
- C(x+ |x|- 1 2 (x^2+1 )+ ⁻¹(x)+c )
- D(x- |x|+ 1 2 (x^2+1 )- ⁻¹(x)+c )
Correct answer
D. (x- |x|+ 1 2 (x^2+1 )- ⁻¹(x)+c )
Step-by-step solution
( aligned & I= ( x^3-1 x^3+x ) d x= (1- x+1 x^3+x ) d x & 1 d x- (x+1) x^3+x d x=x- (x+1) x (x^2+1 ) d x aligned ) Now, ( x+1 x (x^2+1 ) = A x + B x+C x^2+1 ) ( (x+1)=A (x^2+1 )+(B x+C) x ) [using partial fractions] ( (x+1)=(A+B) x^2+C x+A ) On comparing (A+B=0, C=1, A=1 ) ( array lc & B=-1 & I=x- 1 x d x- (1-x) x^2+1 d x & I=x- |x|- 1 x^2+1 d x+ 1 2 2 x x^2+1 d x & I=x- |x|- ⁻¹ x+ 1 2 (x^2+1 )+C array )