Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20265 AprilMorning Shift
Exam
JEE Main
Year
2026
Date
5 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Definite Integration

JEE Main 2026 · 5 April · Morning Shift · Mathematics · Definite Integration

The value of the integral ∫_0^∞ ( _e(x))/(x^2 + 4)\,dx is:

The value of the integral ∫0∞log⁡e(x)x2+4 dx\displaystyle\int_0^\infty \dfrac{\log_e(x)}{x^2 + 4}\,dx is:

Options

  1. Aπlog⁡e(2)2\dfrac{\pi\log_e(2)}{2}
  2. Bπlog⁡e(2)4\dfrac{\pi\log_e(2)}{4}
  3. C1+πlog⁡e(2)1 + \pi\log_e(2)
  4. D2+πlog⁡e(2)2 + \pi\log_e(2)

Correct answer

B. πlog⁡e(2)4\dfrac{\pi\log_e(2)}{4}

Step-by-step solution

Let I=∫0∞log⁡e(x)x2+4 dxI = \displaystyle\int_0^\infty \dfrac{\log_e(x)}{x^2 + 4}\,dx

Substitute x=2tan⁡θx = 2 \tan \theta, which gives dx=2sec⁡2θ dθdx = 2 \sec^2 \theta \,d\theta.

The limits of integration change from x=0→θ=0x = 0 \to \theta = 0 to x→∞→θ=π2x \to \infty \to \theta = \dfrac{\pi}{2}.

I=∫0π/2log⁡e(2tan⁡θ)4tan⁡2θ+4(2sec⁡2θ) dθI = \displaystyle\int_0^{\pi/2} \dfrac{\log_e(2 \tan \theta)}{4 \tan^2 \theta + 4} (2 \sec^2 \theta) \,d\theta

I=∫0π/2log⁡e(2tan⁡θ)4sec⁡2θ(2sec⁡2θ) dθI = \displaystyle\int_0^{\pi/2} \dfrac{\log_e(2 \tan \theta)}{4 \sec^2 \theta} (2 \sec^2 \theta) \,d\theta

I=12∫0π/2log⁡e(2tan⁡θ) dθI = \dfrac{1}{2} \displaystyle\int_0^{\pi/2} \log_e(2 \tan \theta) \,d\theta

I=12∫0π/2(log⁡e2+log⁡e(tan⁡θ)) dθI = \dfrac{1}{2} \displaystyle\int_0^{\pi/2} (\log_e 2 + \log_e(\tan \theta)) \,d\theta

I=12log⁡e2∫0π/21 dθ+12∫0π/2log⁡e(tan⁡θ) dθI = \dfrac{1}{2} \log_e 2 \displaystyle\int_0^{\pi/2} 1 \,d\theta + \dfrac{1}{2} \displaystyle\int_0^{\pi/2} \log_e(\tan \theta) \,d\theta

Let I1=∫0π/2log⁡e(tan⁡θ) dθI_1 = \displaystyle\int_0^{\pi/2} \log_e(\tan \theta) \,d\theta

Using the property ∫0af(x) dx=∫0af(a−x) dx\displaystyle\int_0^a f(x)\,dx = \displaystyle\int_0^a f(a-x)\,dx, we get:

I1=∫0π/2log⁡e(tan⁡(π2−θ)) dθ=∫0π/2log⁡e(cot⁡θ) dθI_1 = \displaystyle\int_0^{\pi/2} \log_e\left(\tan\left(\dfrac{\pi}{2} - \theta\right)\right) \,d\theta = \displaystyle\int_0^{\pi/2} \log_e(\cot \theta) \,d\theta

I1=∫0π/2log⁡e(1tan⁡θ) dθ=−∫0π/2log⁡e(tan⁡θ) dθ=−I1I_1 = \displaystyle\int_0^{\pi/2} \log_e\left(\dfrac{1}{\tan \theta}\right) \,d\theta = -\displaystyle\int_0^{\pi/2} \log_e(\tan \theta) \,d\theta = -I_1

2I1=0⇒I1=02I_1 = 0 \Rightarrow I_1 = 0

Substituting I1=0I_1 = 0 back into the equation for II:

I=12log⁡e2[θ]0π/2+0I = \dfrac{1}{2} \log_e 2 \left[\theta\right]_0^{\pi/2} + 0

I=12log⁡e2(π2)=πlog⁡e24I = \dfrac{1}{2} \log_e 2 \left(\dfrac{\pi}{2}\right) = \dfrac{\pi \log_e 2}{4}

Answer: πlog⁡e(2)4\dfrac{\pi\log_e(2)}{4}
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 2026The value of the integral ∫_ 0 ^ 2 x(x^2+x+1) ( x+1 )( x^4+x^2+1 ) \, dx is equal to: JEE Main 2026If ∫_ π/6 ^ π/4 ( (x-(π)/(3)) (x+(π)/(3))+1)dx = α _e( 3 -1) , then 9α^2 is equal to… JEE Main 2026Let A = bmatrix 1 & 3 & -1 2 & 1 & α 0 & 1 & -1 bmatrix be a singular matrix. Let f(x) =… JEE Main 2026Let 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 2026The value of the integral ∫ _ -1 ^ 1 ((x^3 + |x| + 1)/(x^2 + 2|x| + 1)) dx is equal to : JEE Main 2026The value of the integral ∫_ -π/4 ^ π/4 ( 32 ^4 x 1 + e^ x )dx is: JEE Main 2026Let (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 2026Let f: [1, ∞) arrow R be a differentiable function defined as f(x) = ∫_1^x f(t)\,dt +… JEE Main 2026The value of the integral ∫_ π/6 ^ π/3 ((4 - ^2 x)/( ^4 x)) dx is: JEE Main 2026The integral ∫_ 0 ^ 1 ^ -1 (1+x+x^2)dx is equal to: JEE Main 2026Let f be a twice differentiable function such that f(x)=∫_ 0 ^ x (t-x)dt-∫_ 0 ^ x f(t)… JEE Main 2026Let ∫_ -2 ^ 2 (| x| + [x x])\,dx = 2(3 - 2) + β , where [·] is the greatest integer… JEE Main 2026The value of ∫_0^ 20π ( ^4 x + ^4 x) \, dx is equal to: JEE Main 2026Let [·] denote the greatest integer function. Then the value of ∫_0^3 ( e^x + e^ -x [x]!… JEE Main 2026If 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 PYQsJEE Mathematics PYQs All JEE Main PYQs