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.

In case you are preparing for IIT JEE, you may want to try IIT JEE 1983 Mathematics Archives .

The answer is 1111.

1 solution

Rohith M.Athreya
Jan 19, 2017

$\displaystyle \int \frac{(x-1)e^x}{(x+1)^3}dx=\int e^{t-1}(\frac{1}{t^{2}}-\frac{2}{t^{3}})dt=\frac{e^{t-1}}{t^{2}}+c_{1}+c_{2}=f(x)+c$ where $t-1=x$

$\displaystyle f(x)=\frac{e^{x}}{(x+1)^{2}}+\frac{e}{4}$

compare this with $\displaystyle f(x)=∑_{i=0}^{n}\frac{a_{i}e^{x}}{(x+1)^{i}}+d$

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

In case you are preparing for IIT JEE, you may want to try IIT JEE 1983 Mathematics Archives .

The answer is 1111.

1 solution

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