Question

∫ex[(1 – x)/(1 + x²)]²dx = __________ + c 

(a) e^x(1 + x²) 

(b) [e^x/(1 + x²)] 

(c) e^x[(1 – x)/(1 + x²)] 

(d) e^x(1 – x²)

Answer 1

The correct option (b) [e^x/(1 + x²)] 

Explanation:

e^x[(1 – x)/(1 + x²)]² = e^x[(1 – 2x + x²)/(1 + x²)²] 

= e^x[{(1 + x²) – 2x}/(1 + x²)²] 

= e^x[{1/(1 + x²)}/{(2x)/(1 + x²)²}] 

This is of the form ex [f(x) + f'(x)] 

whose solution is given by e^xf(x) + c 

∴ I = e^x ∙ [1/(1 + x²)] + c 

∴ I = [e^x/(1 + x²)] + c