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