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

Answer: e^x/(1+x^2) + C. Solution: Expand: ((1-x)^2)/((1+x^2)^2) = (1 - 2x + x^2)/((1+x^2)^2) = (1+x^2)/((1+x^2)^2) - 2x/((1+x^2)^2) = 1/(1+x^2) - 2x/((1+x^2)^2). Let f(x) = 1/(1+x^2), then f'(x) = -2x/((1+x^2)^2). Using e^x [f(x)+f'(x)] dx = e^x f(x) + C, the answer is e^x/(1+x^2) + C.

Calculus

Integrals

Hard

$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =$

$\frac{e^x}{1+x^2} + C$

$\frac{-e^x}{1+x^2} + C$

$\frac{e^x}{(1+x^2)^2} + C$

$\frac{-e^x}{(1+x^2)^2} + C$

✅ Correct Option: 1

Expand: $\frac{(1-x)^2}{(1+x^2)^2} = \frac{1 - 2x + x^2}{(1+x^2)^2} = \frac{1+x^2}{(1+x^2)^2} - \frac{2x}{(1+x^2)^2} = \frac{1}{1+x^2} - \frac{2x}{(1+x^2)^2}$ . Let $f(x) = \frac{1}{1+x^2}$ , then $f'(x) = -\frac{2x}{(1+x^2)^2}$ . Using $\int e^x [f(x)+f'(x)] dx = e^x f(x) + C$ , the answer is $\frac{e^x}{1+x^2} + C$ .

2026: 29 May Shift 2

The value of $\int_0^{\pi/2} \frac{\cos x}{\cos\frac{x}{2}+\sin\frac{x}{2}}\,dx$ is

2022: 4 Aug Shift 1

Match List I with List - II

List - I	List - II
A. An even function	I. $x^2 + \cos x$
B. For an even function, $\int_{-a}^{a} f(x)dx =$	II. 0
C. If $f(2a-x) = -f(x)$ , then $\int_{0}^{2a} f(x)dx =$	III. $2\int_{0}^{a} f(x)dx$
D. An odd function	IV. $x^3 + \sin x$

Choose the correct answer from the options given below:

2022: 16 July Shift 2

Match List I with List II

List I	List II
A. $\int \frac{dx}{x^2 - a^2}$ is equal to	I. $\frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$
B. $\int \frac{dx}{\sqrt{x^2 + a^2}}$ is equal to	II. $\frac{1}{2a}\log\left\lvert \frac{x-a}{x+a}\right\rvert + C$
C. $\int \sqrt{a^2 - x^2} \, dx$ is equal to	III. $\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$
D. $\int \sqrt{a^2 + x^2} \, dx$ is equal to	IV. $\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$

Choose the correct answer from the options given below:

2026: 21 May Shift 1

Value of $\int (e^{2\log x} + e^{x\log 2} + e^{2\log 2})dx$ is

2026: 15 May Shift 1

Value of $\displaystyle\int \dfrac{dx}{\sin^{\frac{3}{5}}x \cdot \cos^{\frac{7}{5}}x}$ is (where c is an arbitrary constant)

2026: 21 May Shift 2

Match the LIST-I with LIST-II

	LIST-I Definite Integrals		LIST-II Values
A.	$\displaystyle\int_{-1}^{1} \vert x \vert \, dx$	I.	$2$
B.	$\displaystyle\int_{-1}^{1} \sin^3 x \cos^4 x \, dx$	II.	$1$
C.	$\displaystyle\int_{-1}^{1} 3\vert x \vert^2 \, dx$	III.	$0$
D.	$\displaystyle\int_{0}^{4} \vert x-1 \vert \, dx$	IV.	$5$

Choose the correct answer from the options given below:

∫ex(1−x1+x2)2dx=\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =∫ex(1+x21−x​)2dx=

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$\int e^x \left(\frac{1-x}{1+x^2}\right)^2 dx =$