Let I=∫exe4x+e2x+1dx,J=e3xe4x+e2x+1then thevalue of I – J equals

I−J=∫exe4x+e2x+1dx−∫e3xe4x+e2x+1dx=∫ex1−e2xe4x+e2x+1dxPut ex=t,exdx=dtI−J=∫1−t2t4+t2+1dt=∫1t2−1t2+1t2+1dt=−∫1−1t2t+1t2−1=∫du1−u2u=t+1t=12log⁡1+u1−u+C=12log⁡ex+e−x+1ex+e−x−1+C=12log⁡e2x+ex+1e2x−ex+1+C.