Evaluate the following integral:
$\int \frac{x^{2}}{x^{4}-x^{2}-12} d x$
$\frac{x^{2}}{x^{4}-x^{2}-12}=\frac{x^{2}}{\left(x^{2}-4\right)\left(x^{2}+3\right)}$
Let, $\frac{\mathrm{x}^{2}}{\left(\mathrm{x}^{2}-4\right)\left(\mathrm{x}^{2}+3\right)}=\frac{\mathrm{A}}{\mathrm{x}-2}+\frac{\mathrm{B}}{\mathrm{x}+2}+\frac{\mathrm{C}}{\mathrm{x}^{2}+3}$
$\Rightarrow x^{2}=A(x+2)\left(x^{2}+3\right)+B(x-2)\left(x^{2}+3\right)+C(x-2)(x+2)$
For, $x=2, A=\frac{1}{7}$
For, $x=-2, B=-\frac{1}{7}$
For, $x=0, C=\frac{3}{7}$
$\therefore \int \frac{\mathrm{x}^{2}}{\mathrm{x}^{4}-\mathrm{x}^{2}-12} \mathrm{dx}=\frac{1}{7} \int \frac{\mathrm{dx}}{\mathrm{x}-2}-\frac{1}{7} \int \frac{\mathrm{dx}}{\mathrm{x}+2}+\frac{3}{7} \int \frac{\mathrm{dx}}{\mathrm{x}^{2}+3}$
$=\frac{1}{7} \log |x-2|-\frac{1}{7} \log |x+2|+\frac{3}{7 \sqrt{3}} \tan ^{-1} \frac{x}{\sqrt{3}}+c$
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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,