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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Definite Integration

NDA 2023 · Mathematics · Definite Integration

Paragraph: Let I_1=∫_0^π (x)/(1+ ^2 x) d x and I_2=∫_0^π (1)/(1+ ^2...

Paragraph: Let I1=∫0πx1+cos⁡2xdxI_1=\int_0^\pi \frac{x}{1+\cos ^2 x} d x and I2=∫0π11+sin⁡2xdxI_2=\int_0^\pi \frac{1}{1+\sin ^2 x} d x
Question: What is the value of 8I128 I_1^2 ?

Options

  1. Aπ\pi
  2. Bπ2\pi^2
  3. Cπ3\pi^3
  4. Dπ4\pi^4

Correct answer

D. π4\pi^4

Step-by-step solution

∵I1=π222∴I12=π48⇒8I12=π4\because \quad I_1=\frac{\pi^2}{2 \sqrt{2}}\therefore \quad I_1^2=\frac{\pi^4}{8} \Rightarrow 8 I_1^2=\pi^4
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