The value of integral I = _3^5 x^2/((x-1)(x-2)) dx is Option 1: 2 - 2 + 4 3 Option 2: 2 + 4 3 Option 3: 3 - 2 + 3 Option 4: 2 - 2 + 3

Answer: 2 - 2 + 4 3. Solution: Split as 1 - 1/(x-1) + 4/(x-2). Integrating gives x -, x-1, + 4, x-2, . Evaluating from 3 to 5: (5 - 4 + 4 3) - 3 = 2 - 2 2 + 2 + 4 3 = 2 - 2 + 4 3.

Calculus

Integrals

Medium

common

The value of integral $I = \int_{3}^{5} \frac{x^2}{(x-1)(x-2)}\,dx$ is

$2 - \log 2 + 4\log 3$

$\log 2 + 4\log 3$

$3 - \log 2 + \log 3$

$2 - \log 2 + \log 3$

✅ Correct Option: 1

Split as $1 - \frac{1}{x-1} + \frac{4}{x-2}$ . Integrating gives $x - \log|x-1| + 4\log|x-2|$ . Evaluating from 3 to 5: $(5 - \log 4 + 4\log 3) - 3 = 2 - 2\log 2 + \log 2 + 4\log 3 = 2 - \log 2 + 4\log 3$ .

2025: 21 May Shift 1

Match List-I with List-II

List-I	List-II
Definite integral	Value
---	---
(A) $\int_{1}^{e} \frac{\log x}{x} dx$	(I) 4
(B) $\int_{-2}^{2} x^3(1 - x^2) dx$	(II) $\frac{1}{2}$
(C) $\int_{1}^{2} x \, dx$	(III) 0
(D) $\int_{-2}^{2} \vert x\vert dx$	(IV) $\frac{3}{2}$

Choose the correct answer from the options given below:

2025: 13 May Shift 2

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

2025: 22 May Shift 2

Match List-I with List-II (where $c$ is an arbitrary constant)

List-I	List-II
(A) $\int \tan x \, dx$	(I) $\log\vert \sec x + \tan x\vert + c$
(B) $\int \cot x \, dx$	(II) $\log\vert \sec x\vert + c$
(C) $\int \sec x \, dx$	(III) $\log\vert \sin x\vert + c$
(D) $\int \cosec x \, dx$	(IV) $\log\vert \cosec x - \cot x\vert + c$

Choose the correct answer from the options given below:

2025: 21 May Shift 2

$\int \frac{\sin x - x\cos x}{x(x + \sin x)}dx =$ (where C is an arbitrary constant)

2026: 19 May Shift 2

$\int \frac{\log x - 3}{(\log x)^4} dx$ is equal to

(where C is an arbitrary constant and $\log_e x = \log x$ )

2026: 30 May Shift 2

If $\int_0^1 (4x^3 + 3x^2 + 2x + k)dx = 0$ , then the value of 'k' is

The value of integral I=∫35x2(x−1)(x−2) dxI = \int_{3}^{5} \frac{x^2}{(x-1)(x-2)}\,dxI=∫35​(x−1)(x−2)x2​dx is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The value of integral $I = \int_{3}^{5} \frac{x^2}{(x-1)(x-2)}\,dx$ is