Evaluate the following integral:
$\int \frac{x^{2}}{1-x^{4}} d x$
Let, $\mathrm{I}=\int \frac{\mathrm{x}^{2}}{1-\mathrm{x}^{4}} \mathrm{dx}$
Let, $\frac{\mathrm{x}^{2}}{1-\mathrm{x}^{4}}=\frac{\mathrm{A}}{1-\mathrm{x}}+\frac{\mathrm{B}}{1+\mathrm{x}}+\frac{\mathrm{C}}{1+\mathrm{x}^{2}}$
$\Rightarrow x^{2}=A(1+x)\left(x^{2}+1\right)+B(1-x)\left(x^{2}+1\right)+c(x+1)(1-x)$
For, $x=1, A=\frac{1}{4}$
For, $\mathrm{x}=-1, \mathrm{~B}=\frac{1}{4}$
For, $x=0, C=-\frac{1}{2}$
$\therefore I=\frac{1}{4} \int \frac{d x}{1-x}+\frac{1}{4} \int \frac{d x}{1+x}-\frac{1}{2} \int \frac{d x}{1+x^{2}}$
$\Rightarrow I=-\frac{1}{4} \log |1-x|+\frac{1}{4} \log |1+x|-\frac{1}{2} \tan ^{-1} x+c$
$\Rightarrow I=\frac{1}{4} \log \left|\frac{1+x}{1-x}\right|-\frac{1}{2} \tan ^{-1} x+c$
Hence, $\int \frac{\mathrm{x}^{2}}{1-\mathrm{x}^{4}} \mathrm{dx}=\frac{1}{4} \log \left|\frac{1+\mathrm{x}}{1-\mathrm{x}}\right|-\frac{1}{2} \tan ^{-1} \mathrm{x}+\mathrm{c}$
- 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,
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,