Question:
Evaluate the following integral:
$\int \frac{3}{(1-x)\left(1+x^{2}\right)} d x$
Solution:
$I=\int \frac{3}{(1-x)\left(1+x^{2}\right)} d x$
$\frac{3}{(1-x)\left(1+x^{2}\right)}=\frac{A}{1-x}+\frac{B x+C}{1+x^{2}}$
$3=A\left(1+x^{2}\right)+(B x+C)(1-x)$
Equating similar terms
$A-B=0$
$B-C=0$
$A+C=3$
Solving
$A=\frac{3}{2}, B=\frac{3}{2}, C=\frac{3}{2}$
Thus,
$I=\frac{3}{2} \int \frac{d x}{1-x}+\frac{3}{2} \int \frac{x d x}{1+x^{2}}+\frac{3}{2} \int \frac{d x}{1+x^{2}}$
$=-\frac{3}{2} \log |1-x|+\frac{3}{2} \log \left|1+x^{2}\right|+\frac{3}{2} \tan ^{-1} x+C$
$I=\frac{3}{4}\left[\log \left|\frac{1+x^{2}}{(1-x)^{2}}\right|+2 \tan ^{-1} x\right]+C$
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- 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,
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