AP EAMCET · Maths · Indefinite Integration
\(\int e^x \frac{x^2+1}{(x+1)^2} d x\) is equal to
- A \(\frac{e^x}{x+1}+C\)
- B \(\frac{-e^x}{x+1}+C\)
- C \(e^x\left(\frac{x-1}{x+1}\right)+C\)
- D \(e^x\left(\frac{x+1}{x-1}\right)+C\)
Answer & Solution
Correct Answer
(C) \(e^x\left(\frac{x-1}{x+1}\right)+C\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \text { Let } I=\int e^x \frac{x^2+1}{(x+1)^2} d x=\int e^x\left(\frac{x^2-1+2}{(x+1)^2}\right) d x \\ & =\int e^x\left(\frac{x^2-1}{(x+1)^2}+\frac{2}{(x+1)^2}\right) d x \\ & =\int e^x\left(\frac{x-1}{x+1}+\frac{2}{(x+1)^2}\right) d x\end{aligned}\) Put…
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