Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20266 AprilMorning Shift
Exam
JEE Main
Year
2026
Date
6 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Definite Integration

JEE Main 2026 · 6 April · Morning Shift · Mathematics · Definite Integration

The value of the integral ∫_ -π/4 ^ π/4 ( 32 ^4 x 1 + e^ x )dx is:

The value of the integral ∫−π/4π/4(32cos⁡4x1+esin⁡x)dx\int_{-\pi/4}^{\pi/4}\left(\dfrac{32\cos^4 x}{1 + e^{\sin x}}\right)dx is:

Options

  1. A4π+24\pi + 2
  2. B3π+83\pi + 8
  3. C3π+43\pi + 4
  4. D4π+34\pi + 3

Correct answer

B. 3π+83\pi + 8

Step-by-step solution

Let I=∫−π/4π/432cos⁡4x1+esin⁡xdxI = \int_{-\pi/4}^{\pi/4} \dfrac{32\cos^4 x}{1 + e^{\sin x}} dx

Using the definite integral property ∫−aaf(x)dx=∫0a(f(x)+f(−x))dx\int_{-a}^{a} f(x) dx = \int_{0}^{a} (f(x) + f(-x)) dx, we get:

I=∫0π/4(32cos⁡4x1+esin⁡x+32cos⁡4(−x)1+esin⁡(−x))dxI = \int_{0}^{\pi/4} \left( \dfrac{32\cos^4 x}{1 + e^{\sin x}} + \dfrac{32\cos^4(-x)}{1 + e^{\sin(-x)}} \right) dx

I=∫0π/4(32cos⁡4x1+esin⁡x+32cos⁡4x1+e−sin⁡x)dxI = \int_{0}^{\pi/4} \left( \dfrac{32\cos^4 x}{1 + e^{\sin x}} + \dfrac{32\cos^4 x}{1 + e^{-\sin x}} \right) dx

I=∫0π/4(32cos⁡4x1+esin⁡x+32cos⁡4x⋅esin⁡xesin⁡x+1)dxI = \int_{0}^{\pi/4} \left( \dfrac{32\cos^4 x}{1 + e^{\sin x}} + \dfrac{32\cos^4 x \cdot e^{\sin x}}{e^{\sin x} + 1} \right) dx

I=∫0π/432cos⁡4x(1+esin⁡x1+esin⁡x)dxI = \int_{0}^{\pi/4} 32\cos^4 x \left( \dfrac{1 + e^{\sin x}}{1 + e^{\sin x}} \right) dx

I=∫0π/432cos⁡4xdxI = \int_{0}^{\pi/4} 32\cos^4 x dx

Using the trigonometric identity cos⁡2x=1+cos⁡2x2\cos^2 x = \dfrac{1 + \cos 2x}{2}, we can write:

cos⁡4x=(1+cos⁡2x2)2=14(1+2cos⁡2x+cos⁡22x)\cos^4 x = \left( \dfrac{1 + \cos 2x}{2} \right)^2 = \dfrac{1}{4} (1 + 2\cos 2x + \cos^2 2x)

cos⁡4x=14(1+2cos⁡2x+1+cos⁡4x2)=38+12cos⁡2x+18cos⁡4x\cos^4 x = \dfrac{1}{4} \left( 1 + 2\cos 2x + \dfrac{1 + \cos 4x}{2} \right) = \dfrac{3}{8} + \dfrac{1}{2}\cos 2x + \dfrac{1}{8}\cos 4x

Substituting this back into the integral:

I=32∫0π/4(38+12cos⁡2x+18cos⁡4x)dxI = 32 \int_{0}^{\pi/4} \left( \dfrac{3}{8} + \dfrac{1}{2}\cos 2x + \dfrac{1}{8}\cos 4x \right) dx

I=∫0π/4(12+16cos⁡2x+4cos⁡4x)dxI = \int_{0}^{\pi/4} (12 + 16\cos 2x + 4\cos 4x) dx

Integrating term by term:

I=[12x+8sin⁡2x+sin⁡4x]0π/4I = \left[ 12x + 8\sin 2x + \sin 4x \right]_{0}^{\pi/4}

I=(12(π4)+8sin⁡(π2)+sin⁡(π))−(0+0+0)I = \left( 12\left(\dfrac{\pi}{4}\right) + 8\sin\left(\dfrac{\pi}{2}\right) + \sin(\pi) \right) - (0 + 0 + 0)

I=3π+8(1)+0=3π+8I = 3\pi + 8(1) + 0 = 3\pi + 8

Answer: 3π+83\pi + 8
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 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 ∫_0^∞ ( _e(x))/(x^2 + 4)\,dx is: 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