CUET · MATHS · PYQ PAPER 2023

\(\int e^x\left(\frac{1-x}{1+x^2}\right)^2\) dx is equal to :

A \(\frac{e^x}{1+x^2} + C\)
B \(-\frac{e^x}{1+x^2}+C\)
C \(\frac{e^x}{(1+x^2)^2} + C\)
D \(-\frac{e^x}{\left(1+x^2\right)^2}+C\)

Verified Solution

Answer & Solution

Correct Answer

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

Step-by-step Solution

Detailed explanation

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

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

The number of solutions of the system of equations
\(\begin{array}{l}\alpha^2 x+\alpha y=-1 \\\alpha x+\alpha^2 y=1\end{array}\)
is infinite. Then \(\alpha\) is:

CUET 2023 Hard

Match List-I with List-II

List - I (Differential Equations)	List - II (Degree/Order)
(A) Degree of the differential equation \(\frac{d^3 y}{d x^3}+2 \log x \cdot y=0\)	(I) 3
(B) Order of the differential equation \(\frac{d^4 y}{d x^4}+\left(\frac{d y}{d x}\right)^4+x y=0\)	(II) 2
(C) Degree of the differential equation \(\left(\frac{d^4 y}{d x^4}\right)^2+\left(\frac{d y}{d x}\right)^3+x^2 y=0\)	(III) 1
(D) Order of the differential equation \(\frac{d^3 y}{d x^3}+y\left(\frac{d y}{d x}\right)^3=0\)	(IV) 4

Choose the correct answer from the options given below:

CUET 2025 Hard

Explore more questions on app

From CUET

Explore more questions on app

List-I	List-II
(A) absolute maximum value	(I) 5
(B) absolute minimum value	(II) 60
(C) point of absolute maxima	(III) 3
(D) point of absolute minima	(IV) 32