Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2015
Exam
NDA
Year
2015
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2015 · Mathematics · Indefinite Integration

What is ∫ d x x^2+a^2 equal to? where c is the constant of...

What is ∫dxx2+a2\int \frac{d x}{\sqrt{x^2+a^2}} equal to? where cc is the constant of integration.

Options

  1. Aln⁡∣x+x2+a2a∣+c\ln \left|\frac{\mathrm{x}+\sqrt{\mathrm{x}^2+a^2}}{a}\right|+c
  2. Bln⁡∣x−x2+a2a∣+c\ln \left|\frac{x-\sqrt{x^2+a^2}}{a}\right|+c
  3. Cln⁡∣x2+x2+a2a∣+c\ln \left|\frac{x^2+\sqrt{x^2+a^2}}{a}\right|+c
  4. DNone of these

Correct answer

A. ln⁡∣x+x2+a2a∣+c\ln \left|\frac{\mathrm{x}+\sqrt{\mathrm{x}^2+a^2}}{a}\right|+c

Step-by-step solution

∫dxx2+a2Let x=atan⁡udx=asec⁡2udu=∫asec⁡2udua2tan⁡2u+a2=a∫sec⁡2udua2(1+tan⁡2u)⇒aa∫sec⁡2udusec⁡u\begin{aligned} & \int \frac{d x}{\sqrt{x^2+a^2}} \\ & \text{Let } x=a \tan u \\ & \quad d x=a \sec ^2 u d u \\ & =\int \frac{a \sec ^2 u d u}{\sqrt{a^2 \tan ^2 u+a^2}} \\ & =a \int \frac{\sec ^2 u d u}{\sqrt{a^2\left(1+\tan ^2 u\right)}} \Rightarrow \frac{a}{a} \int \frac{\sec ^2 u d u}{\sec u}\end{aligned}=∫sec⁡2udusec⁡u=∫sec⁡udu=ln⁡[tan⁡(u)+sec⁡(u)]+c[∵∫sec⁡xdx=tan⁡x]\begin{aligned} & =\int \frac{\sec ^2 u d u}{\sec u}=\int \sec u d u \\ & =\ln [\tan (u)+\sec (u)]+c \quad\left[\because \int \sec x d x=\tan x\right]\end{aligned}=ln⁡[xa2+1+x2a2]+c=ln⁡[xa+a2+x2a]+c=ln⁡[x+x2+a2a]+c\begin{aligned} & =\ln \left[\frac{x}{\sqrt{a^2}}+\sqrt{1+\frac{x^2}{a^2}}\right]+c \\ & =\ln \left[\frac{x}{a}+\frac{\sqrt{a^2+x^2}}{a}\right]+c \\ & =\ln \left[\frac{x+\sqrt{x^2+a^2}}{a}\right]+c \end{aligned}\therefore \quad$ Option (A) is correct.
Practice Indefinite Integration on Quantrex Academy →

More from Indefinite Integration

Passage: Let ∫( θ\,dθ)/((2+ θ)(3+4 θ))=A |2+ θ|+B |3+4 θ| . Question: What is the value… NDA 2026Passage: Let ∫( θ\,dθ)/((2+ θ)(3+4 θ))=A |2+ θ|+B |3+4 θ| . Question: What is the value… NDA 2026What is ∫ dx 10^x + 10^ -x equal to? NDA 2025Consider the following for the two (02) items that follow: Let the function y = (1 - x)^… NDA 2025For the following two (02) items: Let u = ∫ e^x x\, dx and v = ∫ e^x x\, dx . Consider… NDA 2025For the following two (02) items: Let u = ∫ e^x x\, dx and v = ∫ e^x x\, dx . What is u +… NDA 2025Let ∫ d x x+1 - x-1 =α(x+1)^ (3)/(2) +β(x-1)^ (3)/(2) +c What is the value of β ? NDA 2024Paragraph: Let 2 ∫ x^2-1 x^2+1 d x=U(x) V(x)-3 \ U(x)+V(x)\ +c Question: What is U(x)… NDA 2024Paragraph: Let 2 ∫ x^2-1 x^2+1 d x=U(x) V(x)-3 \ U(x)+V(x)\ +c Question: What is… NDA 2024Given that ∫ (3 x+4 x)/(2 x+5 x) d x=(α x)/(29)+(β)/(29) |2 x+5 x|+c What is the value of… NDA 2024Given that ∫ (3 x+4 x)/(2 x+5 x) d x=(α x)/(29)+(β)/(29) |2 x+5 x|+c What is the value of… NDA 2024Let ∫ d x x+1 - x-1 =α(x+1)^ (3)/(2) +β(x-1)^ (3)/(2) +c What is the value of α ? NDA 2024Paragraph: Let f(x)=P e^x+Q e^ 2 x +R e^ 3 x , where P, Q, R are real numbers. Further… NDA 2023Paragraph: Let f(x)=P e^x+Q e^ 2 x +R e^ 3 x , where P, Q, R are real numbers. Further… NDA 2023What is ∫(x^x)^2(1+ x) d x equal to? NDA 2022What is ∫ e^ x (1+ x+x x) d x equal to? NDA 2022 Full Indefinite Integration list Indefinite Integration JEE Maths PYQsJEE Mathematics PYQs All NDA PYQs