Evaluate the following integral:
$\int \frac{1}{(x+1)^{2}\left(x^{2}+1\right)} d x$
$I=\frac{1}{(x+1)^{2}\left(x^{2}+1\right)}$
$\frac{1}{(x+1)^{2}\left(x^{2}+1\right)}=\frac{A}{x+1}+\frac{B}{(x+1)^{2}}+\frac{C x+D}{x^{2}+1}$
$1=A(x+1)\left(x^{2}+1\right)+B\left(x^{2}+1\right)+(C x+D)(x+1)^{2}$
$=A x^{3}+A x^{2}+A x+A+B x^{2}+B+C x^{3}+2 C x^{2}+C x+D x^{2}+2 D+D$
$=(A+C) x^{3}+(A+B+2 C+D) x^{2}+(A+C+2 D) x+(A+B+D)$
Equating constants
$1=A+B+D$
Equating coefficients of $x^{3}$
$0=A+C$
Equating coefficients of $x^{2}$
$0=A+B+2 C+D$
Equating coefficients of $x$
$0=A+C+2 D$
Solving we get
$A=\frac{1}{2} B=\frac{1}{2} C=-\frac{1}{2} D=0$
Thus,
$I=\frac{1}{2} \int \frac{\mathrm{dx}}{\mathrm{x}+1}+\frac{1}{2} \int \frac{\mathrm{dx}}{(\mathrm{x}+1)^{2}}-\frac{1}{2} \int \frac{\mathrm{dx}}{\mathrm{x}^{2}+1}$
$I=\frac{1}{2} \log |\mathrm{x}+1|-\frac{1}{2(\mathrm{x}+1)}-\frac{1}{4} \log \left|\mathrm{x}^{2}+1\right|+\mathrm{C}$
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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,