Evaluate the following integral:
$\int \frac{x^{2}}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)} d x$
Let, $I=\int \frac{x^{2}}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)} d x$
Let $x^{2}=y$
Thus, $\frac{\mathrm{x}^{2}}{\left(\mathrm{x}^{2}+\mathrm{a}^{2}\right)\left(\mathrm{x}^{2}+\mathrm{b}^{2}\right)}=\frac{\mathrm{y}}{\left(\mathrm{y}+\mathrm{a}^{2}\right)\left(\mathrm{y}+\mathrm{b}^{2}\right)}$
Now, let $\frac{y}{\left(y+a^{2}\right)\left(y+b^{2}\right)}=\frac{A}{y+a^{2}}+\frac{B}{y+b^{2}}$
$\Rightarrow y=A\left(y+b^{2}\right)+B\left(y+a^{2}\right)$
$\Rightarrow y=y(A+B)+\left(A b^{2}+B a^{2}\right)$
Equating the coefficients, we get,
$A+B=1$, and $A b^{2}+B a^{2}=0$
On solving we get, $A=-\frac{a^{2}}{b^{2}-a^{2}}$ $B=\frac{b^{2}}{b^{2}-a^{2}}$
$\therefore I=-\frac{a^{2}}{b^{2}-a^{2}} \int \frac{d x}{x^{2}+a^{2}}+\frac{b^{2}}{b^{2}-a^{2}} \int \frac{d x}{x^{2}+b^{2}}$
$\Rightarrow I=\frac{b}{b^{2}-a^{2}} \tan ^{-1}\left(\frac{x}{b}\right)-\frac{a}{b^{2}-a^{2}} \tan ^{-1}\left(\frac{x}{a}\right)+c$
Thus, $\int \frac{\mathrm{x}^{2}}{\left(\mathrm{x}^{2}+\mathrm{a}^{2}\right)\left(\mathrm{x}^{2}+\mathrm{b}^{2}\right)} \mathrm{dx}=\frac{\mathrm{b}}{\mathrm{b}^{2}-\mathrm{a}^{2}} \tan ^{-1}\left(\frac{\mathrm{x}}{\mathrm{b}}\right)-\frac{\mathrm{a}}{\mathrm{b}^{2}-\mathrm{a}^{2}} \tan ^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)+\mathrm{c}$ght)+\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,