Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2020
Exam
NDA
Year
2020
Subject
Mathematics
Chapter
Definite Integration

NDA 2020 · Mathematics · Definite Integration

What is the value of ∫_0^ π / 4 ( ^3 x+ x) d x ?

What is the value of ∫0π/4(tan⁡3x+tan⁡x)dx\int_0^{\pi / 4}\left(\tan ^3 x+\tan x\right) d x ?

Options

  1. A14\frac{1}{4}
  2. B12\frac{1}{2}
  3. C1
  4. D2

Correct answer

B. 12\frac{1}{2}

Step-by-step solution

I=∫0π4(tan⁡3x+tan⁡x)⋅dx=∫0π4tan⁡3x⋅dx+∫0π4tan⁡x⋅dx\begin{aligned} I & =\int_0^{\frac{\pi}{4}}\left(\tan ^3 x+\tan x\right) \cdot d x \\ & =\int_0^{\frac{\pi}{4}} \tan ^3 x \cdot d x+\int_0^{\frac{\pi}{4}} \tan x \cdot d x\end{aligned}=∫0π4tan⁡x⋅tan⁡2x⋅dx+∫0π4tan⁡x⋅dx=∫0π4tan⁡x⋅(sec⁡2x−1)⋅dx+∫0π4tan⁡x⋅dx\begin{aligned} & =\int_0^{\frac{\pi}{4}} \tan x \cdot \tan ^2 x \cdot d x+\int_0^{\frac{\pi}{4}} \tan x \cdot d x \\ = & \int_0^{\frac{\pi}{4}} \tan x \cdot\left(\sec ^2 x-1\right) \cdot d x+\int_0^{\frac{\pi}{4}} \tan x \cdot d x\end{aligned}=∫0π4tan⁡x⋅sec⁡2x⋅dx−∫0π4tan⁡x⋅dx+∫0π4tan⁡x⋅dx=∫0π4tan⁡x⋅sec⁡2x⋅dx\begin{aligned} & =\int_0^{\frac{\pi}{4}} \tan x \cdot \sec ^2 x \cdot d x-\int_0^{\frac{\pi}{4}} \tan x \cdot d x+\int_0^{\frac{\pi}{4}} \tan x \cdot d x \\ & =\int_0^{\frac{\pi}{4}} \tan x \cdot \sec ^2 x \cdot d x\end{aligned} Let tan⁡x=z,dz=sec⁡2x⋅dx∴∫tan⁡x⋅sec⁡2x⋅dx=∫z⋅dz=z22+C\begin{aligned} & \text{ Let } \tan x=z, d z=\sec ^2 x \cdot d x \\ & \therefore \int \tan x \cdot \sec ^2 x \cdot d x=\int z \cdot d z=\frac{z^2}{2}+C \end{aligned}wherewhereC=constantconstant\therefore I=\left[\frac{\tan ^2 x}{2}\right]_0^{\frac{\pi}{4}}=\frac{1}{2}-0=\frac{1}{2}$.
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