_1^5, x - 2, dx = Option 1: 5 Option 2: 4 Option 3: 9 Option 4: 1

Answer: 5. Solution: Split at x = 2: _1^2 (2 - x) dx + _2^5 (x - 2) dx = [ 2x - x^2/2 ]_1^2 + [ x^2/2 - 2x ]_2^5 = (4 - 2) - (2 - 1/2) + (25/2 - 10) - (2 - 4) = 2 - 3/2 + 5/2 + 2 = 5.

Calculus

Integrals

Easy

$\int_{1}^{5} |x - 2| \, dx =$

✅ Correct Option: 1

Split at $x = 2$ :

$\int_{1}^{2} (2 - x) \, dx + \int_{2}^{5} (x - 2) \, dx$

$= \left[ 2x - \frac{x^2}{2} \right]_{1}^{2} + \left[ \frac{x^2}{2} - 2x \right]_{2}^{5}$

$= (4 - 2) - (2 - \frac{1}{2}) + (\frac{25}{2} - 10) - (2 - 4)$

$= 2 - \frac{3}{2} + \frac{5}{2} + 2 = 5$ .

2025: 27 May Shift 1

$\int e^{-x}(\cot x + \cosec^2 x)dx =$

2026: 12 May Shift 2

Match the rational functions with the correct form of the partial fractions.

[ $p,q,r,a,b,c,A,B,C$ are arbitrary constants]

Match List-I with List-II

List-I (Rational Function)	List-II (Partial Fraction)
(A) $\dfrac{px^2+qx+r}{(x-a)(x^2+bx+c)}$	(I) $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}+\dfrac{C}{x-b}$
(B) $\dfrac{px^2+qx+r}{(x-a)^2(x-b)}, a\neq b$	(II) $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}$
(C) $\dfrac{px^2+qx+r}{(x-a)(x-b)(x-c)}, a\neq b\neq c$	(III) $\dfrac{A}{x-a}+\dfrac{Bx+C}{x^2+bx+c}$
(D) $\dfrac{px+q}{(x-a)^2}$	(IV) $\dfrac{A}{x-a}+\dfrac{B}{x-b}+\dfrac{C}{x-c}$

Choose the correct answer from the options given below:

2026: 31 May Shift 2

Match the LIST-I with LIST-II (where $[x]$ denotes greatest integer function)

	LIST-I Integral		LIST-II Values
A.	$\displaystyle\int_{0}^{2} [x]\, dx$	I.	$0$
B.	$\displaystyle\int_{0}^{1.5} [x^2]\, dx$	II.	$1$
C.	$\displaystyle\int_{-\pi/4}^{\pi/4} \sin x \, dx$	III.	$\sqrt{2}$
D.	$\displaystyle\int_{-\pi/4}^{\pi/4} \cos x \, dx$	IV.	$2-\sqrt{2}$

Choose the correct answer from the options given below:

2025: 29 May Shift 2

$\int (e^{x\log a} + e^{a\log x}) dx$ is equal to (where $a > 1$ )

2025: 13 May Shift 1

The value of which of the following integrals is zero?

(A) $\int_0^1 x dx$

(B) $\int_{-1}^1 x dx$

(D) $\int_0^1 \log\left(\frac{x}{1-x}\right) dx$

Choose the correct answer from the options given below:

2026: 14 May Shift 2

If C is an arbitrary constant, then

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$\displaystyle\int a^x \, dx$	I.	$\sin^{-1}\dfrac{x}{a} + C$
B.	$\displaystyle\int \dfrac{dx}{\sqrt{a^2 - x^2}}$	II.	$\log_e \vert x + \sqrt{x^2 - a^2} \vert + C$
C.	$\displaystyle\int \dfrac{dx}{\sqrt{a^2 + x^2}}$	III.	$\dfrac{a^x}{\log_e a} + C$
D.	$\displaystyle\int \dfrac{dx}{\sqrt{x^2 - a^2}}$	IV.	$\log_e \vert x + \sqrt{a^2 + x^2} \vert + C$

Choose the correct answer from the options given below:

∫15∣x−2∣ dx=\int_{1}^{5} |x - 2| \, dx =∫15​∣x−2∣dx=

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

$\int_{1}^{5} |x - 2| \, dx =$