# IIT JEE 1983 - 'Adapted' - Subjective to Multi-Correct Q5

Calculus Level 5

Let $\int \frac{(x-1)e^x}{(x+1)^3}dx=f(x)+c$ and $f(x)=\displaystyle\sum_{i=0}^n\frac{a_ie^x}{(x+1)^i}+d$ with all $$a_i=0$$ for $$i\geq n$$ and $$f(1)=\dfrac{e}{2}$$. Then which of the following is/are correct options?

• A: $$a_0=0$$
• B: $$a_1=0$$
• C: $$a_2=1$$
• D: $$f(0)$$ is irrational

Enter your answer as a 4 digit string of 1s and 9s, using 1 for correct option, 9 for wrong. For example, 1199 indicates A and B are correct, C and D are incorrect. None, one or all may also be correct.

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