Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Definite Integration

NDA 2021 · Mathematics · Definite Integration

What is ∫_0^π ( (x)/(2)) d x equal to?

What is ∫0πln⁡(tan⁡x2)dx\int_0^\pi \ln \left(\tan \frac{x}{2}\right) d x equal to?

Options

  1. A0
  2. B12\frac{1}{2}
  3. C1
  4. D2

Correct answer

A. 0

Step-by-step solution

I=∫0πln⁡(tan⁡x2)dx...(i)I=∫0πln⁡(tan⁡(π2−x2))dx=∫0πln⁡(cot⁡x2)dx...(ii)\begin{aligned} \mathrm{I} & =\int_0^\pi \ln \left(\tan \frac{x}{2}\right) d x \qquad \qquad...\mathrm{(i)}\\ \mathrm{I} & =\int_0^\pi \ln \left(\tan \left(\frac{\pi}{2}-\frac{x}{2}\right)\right) d x \\ & =\int_0^\pi \ln \left(\cot \frac{x}{2}\right) d x \qquad \qquad...\mathrm{(ii)} \end{aligned} Adding equation (1) and (2), we get 22\mathrm{I}=∫0πln⁡(tan⁡x2)dx+∫0πln⁡(cot⁡x2)dx=\int_0^\pi \ln \left(\tan \frac{x}{2}\right) d x+\int_0^\pi \ln \left(\cot \frac{x}{2}\right) d x2I=∫0πln⁡(tan⁡x2cot⁡x2)dx2I=∫0πln⁡(tan⁡x2×1tan⁡x2)dxI=12∫0πln⁡(1)dx(∴ln⁡1=0)∴I=0\begin{aligned} 2 \mathrm{I} & =\int_0^\pi \ln \left(\tan \frac{x}{2} \cot \frac{x}{2}\right) d x \\ 2 \mathrm{I} & =\int_0^\pi \ln \left(\tan \frac{x}{2} \times \frac{1}{\tan \frac{x}{2}}\right) d x \\ \mathrm{I} & =\frac{1}{2} \int_0^\pi \ln (1) d x \quad(\therefore \ln 1=0) \\ & \therefore \mathrm{I}=0\end{aligned}$$
Practice Definite Integration on Quantrex Academy →

More from Definite Integration

Passage: Let f(x)= x and g(x)-f(x)=f(4-x) . Question: What is ∫_0^4(f(x))/(g(x))\,dx… NDA 2026Passage: Let ∫_0^ (π)/(2) (a x+b x)/((a+b)( x+ x))\,dx=k . Question: What is the value of… NDA 2026Passage: Let f(x)= x and g(x)-f(x)=f(4-x) . Question: What is ∫_0^4(f(4-x))/(g(4-x))\,dx… NDA 2026Passage: Let ∫_0^ (π)/(2) (a x+b x)/((a+b)( x+ x))\,dx=k . Question: What is ∫_0^ (π)/(2)… NDA 2026Consider the following for the two (02) items that follow: Let f(x) = [x^2] , where [·]… NDA 2025What is ∫_n^ n+1 (x - [x])\, dx , where [·] is the greatest integer function and n is a… NDA 2025If I_1 = ∫_e^ e^2 (dx)/( x) and I_2 = ∫_1^2 (e^x)/(x)\, dx , then which one of the… NDA 2025Paragraph: Let I=∫_0^ (π)/(2) (f(x))/(g(x)) d x , where f(x)= x and g(x)= x+ x+1 .… NDA 2024Paragraph: Let f(t)= (t+ 1+t^2 ) and g(t)= (f(t)) . Question: What is ∫_ -π ^π g(t) d t… NDA 2024Paragraph: Let f(x)=|x^2-x-2| Question: What is ∫_0^2 f(x) d x equal to? NDA 2024What is ∫_ -1 ^1(3 x- 3 x) ^2 x d x equal to? NDA 2024What is ∫_0^ (π)/(2) (a+ x)/(2 a+ x+ x) d x equal to? NDA 2024Let p=∫_a^b f(x) d x and q=∫_a^b|f(x)| d x . If f(x)=e^ -x , then which one of the… NDA 2024Let f(x)=|x-1|, g(x)=[x] and h(x)=f(x) g(x) where [.] is greatest integer function. What… NDA 2024Let f(x)=|x-1|, g(x)=[x] and h(x)=f(x) g(x) where [.] is greatest integer function. What… NDA 2024Let 3 f(x)+f((1)/(x))=(1)/(x)+1 What is 8 ∫_1^2 f(x) d x equal to? NDA 2024 Full Definite Integration list Definite Integration JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs