Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NDA 2014
Exam
NDA
Year
2014
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2014 · Mathematics · Indefinite Integration

Consider ∫ x ^ -1 x d x=A(x^2+1) ^ -1 x+B x+C , where C is the...

Consider ∫xtan⁡−1xdx=A(x2+1)tan⁡−1x+Bx+C\int x \tan ^{-1} x d x=\mathrm{A}\left(x^2+1\right) \tan ^{-1} x+\mathrm{B} x+\mathrm{C}, where C is the constant of integration. What is the value of A ?

Options

  1. A1
  2. B12\frac{1}{2}
  3. C−12-\frac{1}{2}
  4. D14\frac{1}{4}

Correct answer

B. 12\frac{1}{2}

Step-by-step solution

∫x⋅tan⁡−1xdx=A(x2+1)tan⁡−1x+Bx+CLet I=∫x⋅tan⁡−1xdx=tan⁡−1x⋅(x22)−∫11+x2⋅x22dx=x22⋅tan⁡−1x−12∫x21+x2dx=x22⋅tan⁡−1x−12∫(1+x2)−11+x2dx=x22⋅tan⁡−1x−12[∫1dx−∫11+x2dx]=x22⋅tan⁡−1x−12[x−tan⁡−1x]+C=x22⋅tan⁡−1x−12x+12tan⁡−1x+C=(x2+12)tan⁡−1x−x2+C\begin{aligned} & \int x \cdot \tan ^{-1} x d x=\mathrm{A}\left(x^2+1\right) \tan ^{-1} x+\mathrm{B} x+\mathrm{C} \\ & \text{Let } \mathrm{I}=\int x \cdot \tan ^{-1} x d x \\ &=\tan ^{-1} x \cdot\left(\frac{x^2}{2}\right)-\int \frac{1}{1+x^2} \cdot \frac{x^2}{2} d x \\ &=\frac{x^2}{2} \cdot \tan ^{-1} x-\frac{1}{2} \int \frac{x^2}{1+x^2} d x \\ &=\frac{x^2}{2} \cdot \tan ^{-1} x-\frac{1}{2} \int \frac{\left(1+x^2\right)-1}{1+x^2} d x \\ &=\frac{x^2}{2} \cdot \tan ^{-1} x-\frac{1}{2}\left[\int 1 d x-\int \frac{1}{1+x^2} d x\right] \\ &=\frac{x^2}{2} \cdot \tan ^{-1} x-\frac{1}{2}\left[x-\tan ^{-1} x\right]+\mathrm{C} \\ &=\frac{x^2}{2} \cdot \tan ^{-1} x-\frac{1}{2} x+\frac{1}{2} \tan ^{-1} x+\mathrm{C} \\ &=\left(\frac{x^2+1}{2}\right) \tan ^{-1} x-\frac{x}{2}+\mathrm{C}\end{aligned} Value of A=12\mathrm{A}=\frac{1}{2}
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