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CUET Mathematics 2026 PYQs22 May Shift 1. Free, no login required.

Calculus

past year questions, CUET Mathematics 2026 22 May Shift 1

Integrals

past year questions, Calculus, CUET Mathematics 2026 22 May Shift 1

Medium

applied

I=∫(log⁡x)2dxI=\int(\log x)^2dx is equal to :(Consider log⁡ex=log⁡x\log_e x=\log x)

Solution

✅ Correct Option: 3

Integrate by parts with (log⁡x)2(\log x)^2 as the first function and 11 as the second: I=x(log⁡x)2−2∫log⁡x dxI=x(\log x)^2-2\int\log x\,dx.

Since ∫log⁡x dx=xlog⁡x−x\int\log x\,dx=x\log x-x, we get I=x(log⁡x)2−2xlog⁡x+2x+CI=x(\log x)^2-2x\log x+2x+C.

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If C is an arbitrary constant, then

Match the LIST-I with LIST-II

LIST-ILIST-II
A.∫ax dx\displaystyle\int a^x \, dxI.sin⁡−1xa+C\sin^{-1}\dfrac{x}{a} + C
B.∫dxa2−x2\displaystyle\int \dfrac{dx}{\sqrt{a^2 - x^2}}II.log⁡e∣x+x2−a2∣+C\log_e \vert x + \sqrt{x^2 - a^2} \vert + C
C.∫dxa2+x2\displaystyle\int \dfrac{dx}{\sqrt{a^2 + x^2}}III.axlog⁡ea+C\dfrac{a^x}{\log_e a} + C
D.∫dxx2−a2\displaystyle\int \dfrac{dx}{\sqrt{x^2 - a^2}}IV.log⁡e∣x+a2+x2∣+C\log_e \vert x + \sqrt{a^2 + x^2} \vert + C

Choose the correct answer from the options given below:

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