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JEE Main 20244 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
4 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Definite Integration

JEE Main 2024 · 4 April · Evening Shift · Mathematics · Definite Integration

If the value of the integral ∫_ -1 ^1 ( α x)/(1+3^x) d x is (2)/(π)...

If the value of the integral ∫−11cos⁡αx1+3xdx\int_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x is 2π\frac{2}{\pi}. Then, a value of α\alpha is

Options

  1. Aπ3\frac{\pi}{3}
  2. Bπ6\frac{\pi}{6}
  3. Cπ4\frac{\pi}{4}
  4. Dπ2\frac{\pi}{2}

Correct answer

D. π2\frac{\pi}{2}

Step-by-step solution

Let I=∫−1+1cos⁡αx1+3xdxI=\int_{-1}^{+1} \frac{\cos \alpha x}{1+3^x} d x ...(I) I=∫−1+1cos⁡αx1+3−xdxI=\int_{-1}^{+1} \frac{\cos \alpha x}{1+3^{-x}} d x (using⁡∫abf(x)dx=∫abf(a+b−x)dx)\left(u \operatorname{sing} \int_a^b f(x) d x=\int_a^b f(a+b-x) d x\right) ...(II) Add (1) and (II) 2I=∫−1+1cos⁡(αx)dx=2∫01cos⁡(αx)dxI=sin⁡αα=2π( given )∴α=π2\begin{aligned} & 2 I=\int_{-1}^{+1} \cos (\alpha x) d x=2 \int_0^1 \cos (\alpha x) d x \\ & I=\frac{\sin \alpha}{\alpha}=\frac{2}{\pi}(\text{ given }) \\ & \therefore \alpha=\frac{\pi}{2} \end{aligned}
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