_0^1 dx/(x^2 + x + 1) Option 1: ()/3√3 Option 2: ()/2√3 Option 3: ()/6√3 Option 4: ()/√3

Answer: ()/3√3. Solution: Complete the square: x^2+x+1 = (x+1/2)^2 + 3/4. Then _0^1 dx/((x+1/2)^2 + (√3/2)^2) = 2/√3 ^(-1)((2x+1)/√3)|_0^1 = 2/√3[^(-1)√3 - ^(-1)(1/√3)] = 2/√3(()/3 - ()/6) = ()/3√3.

Calculus

Integrals

Medium

$\int_0^1 \frac{dx}{x^2 + x + 1}$

$\frac{\pi}{3\sqrt{3}}$

$\frac{\pi}{2\sqrt{3}}$

$\frac{\pi}{6\sqrt{3}}$

$\frac{\pi}{\sqrt{3}}$

✅ Correct Option: 1

Complete the square: $x^2+x+1 = (x+1/2)^2 + 3/4$ . Then $\int_0^1 \frac{dx}{(x+1/2)^2 + (\sqrt{3}/2)^2} = \frac{2}{\sqrt{3}} \tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)\Big|_0^1 = \frac{2}{\sqrt{3}}[\tan^{-1}\sqrt{3} - \tan^{-1}(1/\sqrt{3})] = \frac{2}{\sqrt{3}}(\frac{\pi}{3} - \frac{\pi}{6}) = \frac{\pi}{3\sqrt{3}}$ .

2025: 21 May Shift 1

The value of $\int_0^{\pi/2} \log_e \left(\frac{5 + 2 \sin x}{5 + 2 \cos x}\right) dx$ is

2023: 30 May Shift 3

$\int e^x \sec x (1 + \tan x) dx$ equals :

2026: 30 May Shift 2

The value of integral $\int_{-\pi/2}^{\pi/2} \frac{1}{1+e^{\sin x}}dx$ is

2026: 12 May Shift 2

The value of $\displaystyle\int_{-1}^{1}|x^4-x|\,dx$ is equal to:

2022: 16 July Shift 2

The value of $\int_{-3}^{3} \log_e\left(\frac{4+x}{4-x}\right) dx$ is

2026: 18 May Shift 1

Match List-I with List-II

List-I	List-II
(A) $\displaystyle\int_{\pi/6}^{\pi/3} \cos x \, dx$	(I) $0$
(B) $\displaystyle\int_{-\pi}^{\pi} (\sin^{61}x + x^{123}) \, dx$	(II) $4$
(C) $\displaystyle\int_{0}^{\pi/2} \dfrac{1}{1+\tan^3 x} \, dx$	(III) $\dfrac{\sqrt{3}-1}{2}$
(D) $\displaystyle\int_{-2}^{2} (2 - \left\vert x \right\vert) \, dx$	(IV) $\dfrac{\pi}{4}$

Choose the correct answer from the options given below:

∫01dxx2+x+1\int_0^1 \frac{dx}{x^2 + x + 1}∫01​x2+x+1dx​

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

$\int_0^1 \frac{dx}{x^2 + x + 1}$