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Questions

Let $I = \int \frac{e^{x}}{e^{4 x} + e^{2 x} + 1} d x, J = \frac{e^{3 x}}{e^{4 x} + e^{2 x} + 1}$ then the
value of $I - J$ equals

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a

12log⁡e4x−e2x+1e4x+e2x+1+C

b

12log⁡e2x+ex+1e2x−ex+1+C

c

12log⁡e2x−ex+1e2x+ex+1+C

d

12log⁡e4x+e2x+1e4x−e2x+1+C

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detailed solution

Correct option is B

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.

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