Evaluate the following integrals:
$\int \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x$
Let $I=\int \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x$
$x=\tan \theta \Rightarrow d x=\sec ^{2} \theta d \theta$
$\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)=\sin ^{-1}\left(\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right)=\sin ^{-1}(\sin 2 \theta)=2 \theta$
$\int \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) d x=\int 2 \theta \sec ^{2} \theta d \theta$
Using integration by parts,
$=2\left(\theta \int \sec ^{2} \theta d \theta-\int \frac{d}{d \theta} \theta \int \sec ^{2} \theta d \theta\right)$
$=2\left(\theta \tan \theta-\int \tan \theta d \theta\right)$
We know that, $\int \tan \theta d \theta=\log |\cos \theta|$
$=2(\theta \tan \theta-\log |\cos \theta|)+\mathrm{c}$\
$=2\left[x \tan ^{-1} x+\log \left|\frac{1}{\sqrt{1+x^{2}}}\right|\right]+c$
$=2 x \tan ^{-1} x+2 \log \left|\left(1+x^{2}\right)^{\frac{1}{2}}\right|+c$
$=2 x \tan ^{-1} x+2\left[\frac{1}{2} \log (1+x)^{2}\right]+c$
$=2 x \tan ^{-1} x+\log (1+x)^{2}+c$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,