The value of the integral ∫_ 0 ^ 2 x(x^2+x+1) ( x+1 )( x^4+x^2+1 ) \, dx is equal to:

Correct answer: C. (2)/(3) _e(3+2 2 ) Let I = ∫_ 0 ^ 2 x(x^2 + x + 1) x + 1 \, x^4 + x^2 + 1 \, dx . Using the identity x^4 + x^2 + 1 = (x^2 + x + 1)(x^2 - x + 1) : I = ∫_ 0 ^ 2 x \, x^2 + x + 1 x + 1 \, (x^2 + x + 1)(x^2 - x + 1) \, dx Cancelling x^2 + x + 1 : I = ∫_ 0 ^ 2 x (x + 1)(x^2 - x + 1) \, dx Using (x + 1)(x^2 - x + 1) = x^3 + 1 : I = ∫_ 0 ^ 2 x x^3 + 1 \, dx Substitution: Let t = x^ 3/2 , so dt = (3)/(2)x^ 1/2 \, dx x \, dx = (2)/(3)\, dt . Also, t^2 = x^3 , so x^3 + 1 = t^2 + 1 . Changing limits: x = 0 t = 0 x = 2 t = 2^ 3/2 = 2 2 Therefore: I = (2)/(3) ∫_ 0 ^ 2 2 dt t^2 + 1 Using ∫ dt t^2 + a^2 = _e|t + t^2 + a^2 | + C : I = (2)/(3)[ _e(t + t^2 + 1 )]_ 0 ^ 2 2 I = (2)/(3)[ _e(2 2 + 8 + 1 ) - _e(1)] I = (2)/(3) _e(2 2 + 3) I = (2)/(3) _e(3 + 2 2 ) Hence, the correct option is (3)\ (2)/(3) _e(3 + 2 2 ) .

JEE Main 20268 AprilEvening Shift

Exam: JEE Main
Year: 2026
Date: 8 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Definite Integration

JEE Main 2026 · 8 April · Evening Shift · Mathematics · Definite Integration

The value of the integral ∫_ 0 ^ 2 x(x^2+x+1) ( x+1 )( x^4+x^2+1 )...

The value of the integral

\displaystyle\int_{0}^{2} \dfrac{\sqrt{x(x^2+x+1)}}{(\sqrt{x+1})(\sqrt{x^4+x^2+1})} \, dx

is equal to:

Options

A $\dfrac{1}{3}\log_e(3-2\sqrt{2})$
B $\dfrac{2}{3}\log_e(4+\sqrt{2})$
C $\dfrac{2}{3}\log_e(3+2\sqrt{2})$
D $\dfrac{1}{3}\log_e(1+6\sqrt{2})$

Correct answer

C. $\dfrac{2}{3}\log_e(3+2\sqrt{2})$

Step-by-step solution

Let

I = \displaystyle\int_{0}^{2} \dfrac{\sqrt{x(x^2 + x + 1)}}{\sqrt{x + 1}\,\sqrt{x^4 + x^2 + 1}}\, dx

.

Using the identity

x^4 + x^2 + 1 = (x^2 + x + 1)(x^2 - x + 1)

I = \displaystyle\int_{0}^{2} \dfrac{\sqrt{x}\,\sqrt{x^2 + x + 1}}{\sqrt{x + 1}\,\sqrt{(x^2 + x + 1)(x^2 - x + 1)}}\, dx

Cancelling

\sqrt{x^2 + x + 1}

I = \displaystyle\int_{0}^{2} \dfrac{\sqrt{x}}{\sqrt{(x + 1)(x^2 - x + 1)}}\, dx

Using

(x + 1)(x^2 - x + 1) = x^3 + 1

I = \displaystyle\int_{0}^{2} \dfrac{\sqrt{x}}{\sqrt{x^3 + 1}}\, dx

Substitution:
Let

t = x^{3/2}

, so

dt = \dfrac{3}{2}x^{1/2}\, dx \Rightarrow \sqrt{x}\, dx = \dfrac{2}{3}\, dt

.

Also,

t^2 = x^3

, so

x^3 + 1 = t^2 + 1

.

Changing limits:

x = 0 \Rightarrow t = 0

x = 2 \Rightarrow t = 2^{3/2} = 2\sqrt{2}

Therefore:

I = \dfrac{2}{3}\displaystyle\int_{0}^{2\sqrt{2}} \dfrac{dt}{\sqrt{t^2 + 1}}

Using

\displaystyle\int \dfrac{dt}{\sqrt{t^2 + a^2}} = \log_e\left|t + \sqrt{t^2 + a^2}\right| + C

I = \dfrac{2}{3}\left[\log_e\left(t + \sqrt{t^2 + 1}\right)\right]_{0}^{2\sqrt{2}}

I = \dfrac{2}{3}\left[\log_e\left(2\sqrt{2} + \sqrt{8 + 1}\right) - \log_e(1)\right]

I = \dfrac{2}{3}\log_e\left(2\sqrt{2} + 3\right)

I = \dfrac{2}{3}\log_e\left(3 + 2\sqrt{2}\right)

Hence, the correct option is

(3)\ \dfrac{2}{3}\log_e\left(3 + 2\sqrt{2}\right)

Practice Definite Integration on Quantrex Academy →

More from Definite Integration

The value of the definite integral ∫_ 0 ^ 2 1 3^ (x+3)/(2) \, dx is JEE Advanced 2026 If ∫_ π/6 ^ π/4 ( (x-(π)/(3)) (x+(π)/(3))+1)dx = α _e( 3 -1) , then 9α^2 is equal to… JEE Main 2026 Let A = bmatrix 1 & 3 & -1 2 & 1 & α 0 & 1 & -1 bmatrix be a singular matrix. Let f(x) =… JEE Main 2026 Let f: ℝ arrow ℝ be such that f(xy) = f(x)f(y) , for all x, y ∈ ℝ and f(0) ≠ 0 . Let g… JEE Main 2026 The value of the integral ∫ _ -1 ^ 1 ((x^3 + |x| + 1)/(x^2 + 2|x| + 1)) dx is equal to : JEE Main 2026 The value of the integral ∫_ -π/4 ^ π/4 ( 32 ^4 x 1 + e^ x )dx is: JEE Main 2026 Let (2^ 1-a + 2^ 1+a ) , f(a) , (3^a + 3^ -a ) be in A.P. and α be the minimum value of… JEE Main 2026 Let f: [1, ∞) arrow R be a differentiable function defined as f(x) = ∫_1^x f(t)\,dt +… JEE Main 2026 The value of the integral ∫_0^∞ ( _e(x))/(x^2 + 4)\,dx is: JEE Main 2026 The value of the integral ∫_ π/6 ^ π/3 ((4 - ^2 x)/( ^4 x)) dx is: JEE Main 2026 The integral ∫_ 0 ^ 1 ^ -1 (1+x+x^2)dx is equal to: JEE Main 2026 Let f be a twice differentiable function such that f(x)=∫_ 0 ^ x (t-x)dt-∫_ 0 ^ x f(t)… JEE Main 2026 Let ∫_ -2 ^ 2 (| x| + [x x])\,dx = 2(3 - 2) + β , where [·] is the greatest integer… JEE Main 2026 The value of ∫_0^ 20π ( ^4 x + ^4 x) \, dx is equal to: JEE Main 2026 Let [·] denote the greatest integer function. Then the value of ∫_0^3 ( e^x + e^ -x [x]!… JEE Main 2026 If α = ∫_0^ 2 3 _2(x^2 + 4)\,dx + ∫_2^4 2^x - 4 \,dx , then α^2 is equal to _______. JEE Main 2026 Full Definite Integration list Definite Integration JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs