MHT CET · Maths · Indefinite Integration
\(\int \frac{\log \left(x^2+\mathrm{a}^2\right)}{x^2} \mathrm{~d} x=\)
- A \(\frac{-\log \left(x^2+\mathrm{a}^2\right)}{x}+\frac{1}{\mathrm{a}} \tan ^{-1} \frac{x}{\mathrm{a}}+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
- B \(\frac{-\log \left(x^2+\mathrm{a}^2\right)}{x}+\frac{2}{\mathrm{a}} \tan ^{-1} \frac{x}{\mathrm{a}}+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
- C \(\frac{\log \left(x^2+\mathrm{a}^2\right)}{x^2}-\frac{1}{\mathrm{a}} \tan ^{-1} \frac{x}{\mathrm{a}}+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
- D \(\frac{\log \left(x^2+\mathrm{a}^2\right)}{x^2}-\frac{2}{\mathrm{a}} \tan ^{-1} \frac{x}{\mathrm{a}}+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
Answer & Solution
Correct Answer
(B) \(\frac{-\log \left(x^2+\mathrm{a}^2\right)}{x}+\frac{2}{\mathrm{a}} \tan ^{-1} \frac{x}{\mathrm{a}}+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
Step-by-step Solution
Detailed explanation
Let
\(\mathrm{I} =\int \frac{\log \left(x^2+\mathrm{a}^2\right)}{x^2} \mathrm{~d} x \)
\( =\int \log \left(x^2+\mathrm{a}^2\right) \cdot x^{-2} \mathrm{~d} x \)
\( =\log \left(x^2+\mathrm{a}^2\right) \int x^{-2} \mathrm{~d} x \) \(-\int\left\{\frac{\mathrm{d}}{\mathrm{d} x}\left[\log \left(x^2+\mathrm{a}^2\right)\right] \int x^{-2} \mathrm{~d} x\right\} \mathrm{d} x \)
\( =\log \left(x^2+\mathrm{a}^2\right) \cdot\left(-\frac{1}{x}\right)-\int \frac{2 x}{x^2+\mathrm{a}^2} \cdot\left(-\frac{1}{x}\right) \mathrm{d} x \)
\( =-\frac{\log \left(x^2+\mathrm{a}^2\right)}{x}+2 \int \frac{1}{x^2+\mathrm{a}^2} \mathrm{~d} x \)
\( =-\frac{\log \left(x^2+\mathrm{a}^2\right)}{x}+\frac{2}{\mathrm{a}} \tan ^{-1}\left(\frac{x}{\mathrm{a}}\right)+\mathrm{c}\)
\(\mathrm{I} =\int \frac{\log \left(x^2+\mathrm{a}^2\right)}{x^2} \mathrm{~d} x \)
\( =\int \log \left(x^2+\mathrm{a}^2\right) \cdot x^{-2} \mathrm{~d} x \)
\( =\log \left(x^2+\mathrm{a}^2\right) \int x^{-2} \mathrm{~d} x \) \(-\int\left\{\frac{\mathrm{d}}{\mathrm{d} x}\left[\log \left(x^2+\mathrm{a}^2\right)\right] \int x^{-2} \mathrm{~d} x\right\} \mathrm{d} x \)
\( =\log \left(x^2+\mathrm{a}^2\right) \cdot\left(-\frac{1}{x}\right)-\int \frac{2 x}{x^2+\mathrm{a}^2} \cdot\left(-\frac{1}{x}\right) \mathrm{d} x \)
\( =-\frac{\log \left(x^2+\mathrm{a}^2\right)}{x}+2 \int \frac{1}{x^2+\mathrm{a}^2} \mathrm{~d} x \)
\( =-\frac{\log \left(x^2+\mathrm{a}^2\right)}{x}+\frac{2}{\mathrm{a}} \tan ^{-1}\left(\frac{x}{\mathrm{a}}\right)+\mathrm{c}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The equation of the plane through the point \((2,-1,-3)\) and parallel to the lines \(\frac{x-1}{3}=\frac{y+2}{2}=\frac{z}{-4}\) and \(\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}\) isMHT CET 2024 Medium
- The point where the line \(\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z+3}{4}\) meets the plane \(2 x+4 y-z=1\), isMHT CET 2010 Easy
- The maximum value of the function \(a \sin x+\mathrm{b} \cos x\) isMHT CET 2025 Easy
- The perimeter of the triangle whose vertices have the position vectos \(\hat{\imath}+\hat{\jmath}+\hat{\mathrm{k}}, 5 \hat{\imath}+3 \hat{\jmath}-3 \hat{\mathrm{k}}\) and \(2 \hat{\imath}+5 \hat{\jmath}+9 \hat{\mathrm{k}}\) isMHT CET 2020 Medium
- The objective function subject to has minimum value at the pointMHT CET 2017 Hard
- After \(t\) seconds, the acceleration of a particle, which starts from rest and moves in a straight line is \(\left(8-\frac{\mathrm{t}}{5}\right) \mathrm{cm} / \mathrm{s}^2\), then velocity of the particle at the instant, when the acceleration is zero, isMHT CET 2024 Easy
More PYQs from MHT CET
- Calculate the edge length of fcc unit cell if radius of metal atom is 139 pm .MHT CET 2024 Medium
- For the combustion of 1 mole of liquid benzene at \(298 \mathrm{~K}\), the heat of reaction at constant pressure is \(-3268 \mathrm{~kJ} \mathrm{~mol}^{-1}\), what is heat of combustion at constant volume? \(\left(\mathrm{R}=8 \cdot 314 \times 10^{-3} \mathrm{~kJ} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right.\)MHT CET 2020 Hard
- The vector sum of two forces \(\overrightarrow{\mathrm{A}}\) and \(\overrightarrow{\mathrm{B}}\) is perpendicular to their vector difference. Hence forces \(\vec{A}\) and \(\vec{B}\) areMHT CET 2025 Medium
- A particle is performing a linear simple harmonic motion of amplitude ‘A’. When it is midway between its mean and extreme position, the magnitudes of its velocity and acceleration are equal. What is the periodic time of the motion?MHT CET 2019 Medium
- Which of the following compounds when treated with ammoniacal silver nitrate exhibits silver mirror test?MHT CET 2024 Easy
- \(\int \mathrm{e}^{x}\left(\frac{1-x}{1+x^{2}}\right)^{2} \mathrm{~d} x=\)MHT CET 2020 Medium