COMEDK · Maths · 28. Indefinite Integration
\(\int \dfrac{(x-1)}{(x+1)^3} e^x d x=P(x)+c\) then \(P(x)=\)
- A \(-\dfrac{e^x}{x+1}\)
- B \(\dfrac{e^x}{x+1}\)
- C \(\dfrac{e^x}{(x+1)^3}\)
- D \(\dfrac{e^x}{(x+1)^2}\)
Answer & Solution
Correct Answer
(D) \(\dfrac{e^x}{(x+1)^2}\)
Step-by-step Solution
Detailed explanation
The integral is given by \(I = \int \dfrac{x-1}{(x+1)^3} e^x dx\). Rewrite the numerator \(x-1\) as \((x+1) - 2\): \(I = \int \dfrac{(x+1) - 2}{(x+1)^3} e^x dx = \int \left( \dfrac{x+1}{(x+1)^3} - \dfrac{2}{(x+1)^3} \right) e^x dx\)…
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