xe^x/((x+1)^2) dx = Option 1: e^x/(x+1) + c Option 2: e^x/(x-1) + c Option 3: x/(x+1) + c Option 4: x/(x-1) + c

Answer: e^x/(x+1) + c. Solution: Use e^x [f(x)+f'(x)] dx = e^x f(x) + C. Let f(x)=1/(x+1), then f'(x)=-1/((x+1)^2). So f(x)+f'(x)=1/(x+1)-1/((x+1)^2)=x/((x+1)^2). Hence xe^x/((x+1)^2) dx = e^x/(x+1)+C.

Calculus

Integrals

Medium

$\int \frac{xe^x}{(x+1)^2} dx =$

$\frac{e^x}{x+1} + c$

$\frac{e^x}{x-1} + c$

$\frac{x}{x+1} + c$

$\frac{x}{x-1} + c$

✅ Correct Option: 1

Use $\int e^x [f(x)+f'(x)]\,dx = e^x f(x) + C$ .

Let $f(x)=\frac{1}{x+1}$ , then $f'(x)=-\frac{1}{(x+1)^2}$ . So $f(x)+f'(x)=\frac{1}{x+1}-\frac{1}{(x+1)^2}=\frac{x}{(x+1)^2}$ .

Hence $\int \frac{xe^x}{(x+1)^2}\,dx = \frac{e^x}{x+1}+C$ .

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Match List-I with List-II

[.] denotes the greatest integer function.

List-I	List-II
(A) $\int_0^3 [x]dx$	(I) $\frac{1}{2}$
(B) $\int_0^1 [2x]dx$	(II) 1
(C) $\int_0^1 [3x]dx$	(III) $\frac{3}{2}$
(D) $\int_0^1 [4x]dx$	(IV) 3

Choose the correct answer from the options given below:

∫xex(x+1)2dx=\int \frac{xe^x}{(x+1)^2} dx =∫(x+1)2xex​dx=

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$\int \frac{xe^x}{(x+1)^2} dx =$