Manipal MET2012MathematicsIndefinite Integration
Evaluate ( x + x ) d x
Options
- A2 ⁻¹( x - x )+C
- B2 2 ⁻¹ x+C
- C2 ⁻¹ ( x - x 2 )+C
- DNone of the above
Correct answer
C. 2 ⁻¹ ( x - x 2 )+C
Step-by-step solution
Let aligned I & = ( x + x ) d x & = x+1 x d x aligned Put x=t^2 aligned & ^2 x d x=2 t d t & d x= 2 t d t 1+ ^2 t = 2 t 1+t^4 d t & aligned I & = t^2+1 t^2 2 t t^4+1 d t & =2 t^2+1 t^4+1 d t aligned aligned aligned &=2 1+ 1 t^2 t^2+ 1 t^2 -2+2 d t &=2 1+ 1 t^2 (t- 1 t )^2+( 2 )^2 d t &=2 d u u^2+( 2 )^2 & where, aligned u & =t- 1 t d u= (1+ 1 t^2 ) d t I & = 2 2 ⁻¹ ( u 2 )+C & = 2 ⁻¹ ( x - x 2 )+C aligned aligned