Evaluate the following integral:
$\int \frac{2 \mathrm{x}}{\mathrm{x}^{3}-1} \mathrm{dx}$
$I=\int \frac{2 x}{x^{3}-1} d x=\int \frac{2 x}{(x-1)\left(x^{2}+x+1\right)} d x$
$\frac{2 x}{(x-1)\left(x^{2}+x+1\right)}=\frac{A}{x-1}+\frac{B x+C}{x^{2}+x+1}$
$2 x=A\left(x^{2}+x+1\right)+(B x+C)(x-1)$
$=(A+B) x^{2}+(A-B+C) x+(A-C)$
Equating constants,
$A-C=0$
Equating coefficients of $x$
$2=A-B+C$
Equating coefficients of $x^{2}$
$0=A+B$
On solving,
We get
$A=\frac{2}{3} B=-\frac{2}{3} C=\frac{2}{3}$
$I=\frac{2}{3} \int \frac{d x}{x-1}-\frac{2}{3} \int \frac{(x-1) d x}{x^{2}+x+1}$
$=\frac{2}{3} \int \frac{\mathrm{dx}}{\mathrm{x}-1}-\frac{2}{3} \cdot \frac{1}{2} \int \frac{(2 \mathrm{x}-2) \mathrm{dx}}{\mathrm{x}^{2}+\mathrm{x}+1}$
$=\frac{2}{3} \int \frac{\mathrm{dx}}{\mathrm{x}-1}-\frac{1}{3} \int \frac{(2 \mathrm{x}+1) \mathrm{dx}}{\mathrm{x}^{2}+\mathrm{x}+1}+\int \frac{\mathrm{dx}}{\mathrm{x}^{2}+\mathrm{x}+1}$
$=\frac{2}{3} \int \frac{\mathrm{dx}}{\mathrm{x}-1}-\frac{1}{3} \int \frac{(2 \mathrm{x}+1) \mathrm{dx}}{\mathrm{x}^{2}+\mathrm{x}+1}+\int \frac{\mathrm{dx}}{\left(\mathrm{x}+\frac{1}{2}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}$
$=\frac{2}{3} \log |\mathrm{x}-1|-\frac{1}{3} \log \left|\mathrm{x}^{2}+\mathrm{x}+1\right|+\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 \mathrm{x}+1}{\sqrt{3}}\right)+\mathrm{C}$
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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,