NDA 2010
Exam NDA
Year 2010
Subject Mathematics
Chapter Definite Integration NDA 2010 · Mathematics · Definite Integration
What is ∫_ (-π)/(4) ^ (π)/(4) ^3 x d x equal to?
What is
∫ − π 4 π 4 tan 3 x d x \int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} \tan ^3 x d x ∫ 4 − π 4 π tan 3 x d x equal to?
Options A 3 \sqrt{3} 3 B 1 3 \frac{1}{3} 3 1 C 1 2 \frac{1}{2} 2 1 D 0
Step-by-step solution
A = ∫ − π / 4 π / 4 tan x tan 2 x d x = ∫ − π / 4 π / 4 tan x ( sec 2 x − 1 ) d x = ∫ − π / 4 π / 4 tan x ⋅ sec 2 x d x − ∫ − π / 4 π / 4 tan x d x \begin{aligned}
\mathrm{A} & =\int_{-\pi / 4}^{\pi / 4} \tan x \tan ^2 x d x \\
& =\int_{-\pi / 4}^{\pi / 4} \tan x\left(\sec ^2 x-1\right) d x \\
& =\int_{-\pi / 4}^{\pi / 4} \tan x \cdot \sec ^2 x d x-\int_{-\pi / 4}^{\pi / 4} \tan x d x
\end{aligned} A = ∫ − π /4 π /4 tan x tan 2 x d x = ∫ − π /4 π /4 tan x ( sec 2 x − 1 ) d x = ∫ − π /4 π /4 tan x ⋅ sec 2 x d x − ∫ − π /4 π /4 tan x d x
Put
tan x = t \tan x=t tan x = t or
sec 2 x d x = d t \sec ^2 x d x=d t sec 2 x d x = d t A = ∫ − 1 1 t d t − [ log sec x ] − π / 4 π / 4 = [ t 2 2 ] − 1 1 − [ log sec ( π / 4 ) − log sec ( − π / 4 ) ] = 0 − [ log 2 − log 2 ] = 0 \begin{aligned} & A=\int_{-1}^1 t d t-[\log \sec x]_{-\pi / 4}^{\pi / 4} \\ & =\left[\frac{t^2}{2}\right]_{-1}^1-[\log \sec (\pi / 4)-\log \sec (-\pi / 4)] \\ & =0-[\log \sqrt{2}-\log \sqrt{2}]=0\end{aligned} A = ∫ − 1 1 t d t − [ log sec x ] − π /4 π /4 = [ 2 t 2 ] − 1 1 − [ log sec ( π /4 ) − log sec ( − π /4 )] = 0 − [ log 2 − log 2 ] = 0 $$
Practice Definite Integration on Quantrex Academy →
← Previous question What is the value of ∫_1^2 e^x((1)/(x)-(1)/(x^2)) d xNext question → What is ∫_0^ π / 2 ( ^2 x)/( ^3 x+ ^3 x) d x ?
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 2026 Passage: Let ∫_0^ (π)/(2) (a x+b x)/((a+b)( x+ x))\,dx=k . Question: What is the value of… NDA 2026 Passage: 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 2026 Passage: Let ∫_0^ (π)/(2) (a x+b x)/((a+b)( x+ x))\,dx=k . Question: What is ∫_0^ (π)/(2)… NDA 2026 Consider the following for the two (02) items that follow: Let f(x) = [x^2] , where [·]… NDA 2025 What is ∫_n^ n+1 (x - [x])\, dx , where [·] is the greatest integer function and n is a… NDA 2025 If I_1 = ∫_e^ e^2 (dx)/( x) and I_2 = ∫_1^2 (e^x)/(x)\, dx , then which one of the… NDA 2025 Paragraph: Let I=∫_0^ (π)/(2) (f(x))/(g(x)) d x , where f(x)= x and g(x)= x+ x+1 .… NDA 2024 Paragraph: Let f(t)= (t+ 1+t^2 ) and g(t)= (f(t)) . Question: What is ∫_ -π ^π g(t) d t… NDA 2024 Paragraph: Let f(x)=|x^2-x-2| Question: What is ∫_0^2 f(x) d x equal to? NDA 2024 What is ∫_ -1 ^1(3 x- 3 x) ^2 x d x equal to? NDA 2024 What is ∫_0^ (π)/(2) (a+ x)/(2 a+ x+ x) d x equal to? NDA 2024 Let 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 2024 Let f(x)=|x-1|, g(x)=[x] and h(x)=f(x) g(x) where [.] is greatest integer function. What… NDA 2024 Let f(x)=|x-1|, g(x)=[x] and h(x)=f(x) g(x) where [.] is greatest integer function. What… NDA 2024 Let 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 PYQs JEE Mathematics PYQs
All NDA PYQs
