Evaluate the following integrals:
$\int \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) d x$
Let $I=\int \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) d x$
$\mathrm{d} \mathrm{x}=\sec ^{2} \mathrm{t} \mathrm{dt}$
$d x=\sec ^{2} t d t$
$I=\int \cos ^{-1}\left(\frac{1-\tan ^{2} t}{1+\tan ^{2} t}\right) \sec ^{2} t d t$
We know that $\frac{1-\tan ^{2} t}{1+\tan ^{2} t}=\cos 2 t$
$=\int \cos ^{-1}(\cos 2 t) \sec ^{2} t d t$
$=\int 2 t \sec ^{2} t d t$
Using integration by parts,
$=2\left[\mathrm{t} \int \sec ^{2} \mathrm{t} \mathrm{dt}-\int \frac{\mathrm{d}}{\mathrm{dt}} \mathrm{t} \int \mathrm{sec}^{2} \mathrm{t} \mathrm{dt}\right]$
$=2\left[\mathrm{t} \tan \mathrm{t}-\int \tan \mathrm{t} \mathrm{dt}\right]$
$=2[\mathrm{t} \tan \mathrm{t}-\log \sec \mathrm{t}]+\mathrm{c}$
$=2\left[\mathrm{x} \tan ^{-1} \mathrm{x}-\log \left|\sqrt{1+\mathrm{x}^{2}}\right|\right]+\mathrm{c}$
$=2 \mathrm{xtan}^{-1} \mathrm{x}-\log \left|1+\mathrm{x}^{2}\right|+\mathrm{c}$
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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,