Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Definite Integration

NDA 2016 · Mathematics · Definite Integration

Consider the functions f(x)=x g(x) and g(x)= [(1)/(x)] Where [ ·]...

Consider the functions f(x)=xg(x) and g(x)=[1x]f(x)=x g(x) \text{ and } g(x)= \left[\frac{1}{x}\right] Where [[\cdot] is the greatest integer function. What is \int_{13\frac{1}{3} }^{12\frac{1}{2}} g(x) d x equal to?

Options

  1. A16\frac{1}{6}
  2. B13\frac{1}{3}
  3. C518\frac{5}{18}
  4. D536\frac{5}{36}

Correct answer

B. 13\frac{1}{3}

Step-by-step solution

∫1/31/2g(x)dxg(x)={2, if  13<x≤123, if  x=13\begin{aligned} & \int_{1/3}^{1 / 2} \mathrm{ g}(x) \mathrm{d} x \\ & \mathrm{ g}(\mathrm{x})=\left\{\begin{array}{lc} 2, & \text{\text{ if } } \frac{1}{3} < x \leq \frac{1}{2} \\ 3, & \text{\text{ if } } x=\frac{1}{3} \end{array}\right. \end{aligned} As g(x)g(x) is a gretest integer function so value of g(x)g(x) in integral limit will be So ∫1/31/2g(x)dx=∫1/31/22dx\int_{1/3}^{1 / 2} g(x) d x=\int_{1/3}^{1 / 2} 2 d x=2[x]1/31/2=2[12−13]=13\begin{aligned} & =2[x]_{1/3}^{1 / 2} \\ & =2\left[\frac{1}{2}-\frac{1}{3}\right] \\ & =\frac{1}{3} \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