Definite Integration
Substitution method
Grade Class 12
Question:
Let I = ∫ e^x / (e^4x + e^2x + 1) dx, J = ∫ e^-x / (e^-4x + e^-2x + 1) dx. Then, for an arbitrary constant c, the value of J - I equals
(A) 1/2 * ln((e^4x - e^2x + 1) / (e^4x + e^2x + 1)) + c
(B) 1/2 * ln((e^2x + e^x + 1) / (e^2x - e^x + 1)) + c
(C) 1/2 * ln((e^2x - e^x + 1) / (e^2x + e^x + 1)) + c
(D) 1/2 * ln((e^4x + e^x + 1) / (e^4x - e^2x + 1)) + c
Step-by-Step Solution
Key Concept: Simplify J by multiplying numerator and denominator by e^4x
J = \int e^3x / (1 + e^2x + e^4x) dx. Then J - I = \int (e^3x - e^x) / (e^4x + e^2x + 1) dx. Substitute e^x = t.
Correct Answer: C