\1/(_e x) - 1/((_e x)^2)\dx is equal to : (where C is an arbitrary constant) Option 1: x_e x + C Option 2: x/(_e x)+C Option 3: e^x_e x + C Option 4: e^x/(_e x)+C

Answer: x/(_e x)+C. Solution: Observe that d/dx(x/(_e x))=(_e x-1)/((_e x)^2)=1/(_e x)-1/((_e x)^2), exactly the integrand. Hence the integral is x/(_e x)+C.

Calculus

Integrals

Medium

common

$\int\left\{\dfrac{1}{\log_e x} - \dfrac{1}{(\log_e x)^2}\right\}dx$ is equal to : (where $C$ is an arbitrary constant)

$x\log_e x + C$

$\dfrac{x}{\log_e x}+C$

$e^x\log_e x + C$

$\dfrac{e^x}{\log_e x}+C$

✅ Correct Option: 2

Observe that $\dfrac{d}{dx}\left(\dfrac{x}{\log_e x}\right)=\dfrac{\log_e x-1}{(\log_e x)^2}=\dfrac{1}{\log_e x}-\dfrac{1}{(\log_e x)^2}$ , exactly the integrand. Hence the integral is $\dfrac{x}{\log_e x}+C$ .

2026: 18 May Shift 1

Match List-I with List-II

List-I	List-II
Integral	Value of the integral
(A) $\displaystyle\int_{1}^{2} e^{2\log x} \, dx$	(I) $\log 2$
(B) $\displaystyle\int_{-1}^{1} \left\vert x \right\vert \, dx$	(II) $\dfrac{1}{2}$
(C) $\displaystyle\int_{0}^{1} \log e^x \, dx$	(III) $\dfrac{7}{3}$
(D) $\displaystyle\int_{1}^{2} \dfrac{1}{x} \, dx$	(IV) $1$

Choose the correct answer from the options given below:

2022: 30 Aug Shift 1

$\int \sqrt{1 - 49x^2} \, dx$ is equal to

2025: 22 May Shift 1

$\int \left(\frac{1}{\log_e x} - \frac{1}{(\log_e x)^2}\right)dx$ is equal to

2025: 13 May Shift 1

The value of the integral $I = \int_0^1 \frac{1}{\sqrt{1+3\sqrt{x}}} dx$ is

2026: 25 May Shift 1

$\displaystyle\int \dfrac{1}{(x-1)^{3/4}(x+2)^{5/4}}\, dx$ is equal to :

[where C is an arbitrary constant]

2026: 12 May Shift 2

The value of the integral $I=\displaystyle\int\dfrac{e^x(x-2)}{(x+1)^4}dx$ is, [ $C$ is an arbitrary constant]

∫{1log⁡ex−1(log⁡ex)2}dx\int\left\{\dfrac{1}{\log_e x} - \dfrac{1}{(\log_e x)^2}\right\}dx∫{loge​x1​−(loge​x)21​}dx is equal to : (where CCC is an arbitrary constant)

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

$\int\left\{\dfrac{1}{\log_e x} - \dfrac{1}{(\log_e x)^2}\right\}dx$ is equal to : (where $C$ is an arbitrary constant)