Integration Question ?
How do you do this ?
∫ e^(x) dx * [(1 - x)/(1 + x)]²
SOLUTION : Let I = ∫ e^(x) * [(1 - x)/(1 + x)]² dx
=> ∫ e^(x) * (1 - 2x + x²) / (1 + x)² dx
=> ∫ e^(x) * (4 + x² - 2x - 3) / (1 + x)² dx
=> ∫ e^(x) * [ 4 / (1 + x)² + (x² - 2x - 3) / (1 + x)² ] dx
=> ∫ e^(x) * [ 4 / (1 + x)² + (x -3)(x + 1) / (1 + x)² ] dx
=> ∫ e^(x) * [ 4 / (1 + x)² + (x - 3) / (1 + x) ] dx
Now, using the fact, ∫ e^(x) * [ f(x) + f'(x)] dx = e^(x) * f(x) + C
here, f(x) = (x - 3) / (1 + x) and f'(x) = 4 / (1 + x)²
so, I = e^(x) * (x - 3) / (1 + x) + C ==> (Ans)
Tuesday, January 26, 2010
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