Evaluate the following integral:
$\int \frac{x}{\left(x^{2}-a^{2}\right)\left(x^{2}-b^{2}\right)} d x$
Denominator is factorized, so let separate the fraction through partial fraction, hence let
$\frac{\mathrm{x}}{\left(\mathrm{x}^{2}-\mathrm{a}^{2}\right)\left(\mathrm{x}^{2}-\mathrm{b}^{2}\right)}=\frac{\mathrm{Ax}+\mathrm{B}}{\left(\mathrm{x}^{2}-\mathrm{a}^{2}\right)}+\frac{\mathrm{Cx}+\mathrm{D}}{\left(\mathrm{x}^{2}-\mathrm{b}^{2}\right)} \ldots \ldots$ (i)
$\Rightarrow \frac{\mathrm{x}}{\left(\mathrm{x}^{2}-\mathrm{a}^{2}\right)\left(\mathrm{x}^{2}-\mathrm{b}^{2}\right)}=\frac{(\mathrm{Ax}+\mathrm{B})\left(\mathrm{x}^{2}-\mathrm{b}^{2}\right)+(\mathrm{Cx}+\mathrm{D})\left(\mathrm{x}^{2}-\mathrm{a}^{2}\right)}{\left(\mathrm{x}^{2}-\mathrm{a}^{2}\right)\left(\mathrm{x}^{2}-\mathrm{b}^{2}\right)}$
$\Rightarrow x=(A x+B)\left(x^{2}-b^{2}\right)+(C x+D)\left(x^{2}-a^{2}\right)$
$\Rightarrow x=A x^{3}-A b^{2} x+B x^{2}-b^{2} B+C x^{3}-a^{2} C x+D x^{2}-a^{2} D$
$\Rightarrow \mathrm{x}=(\mathrm{A}+\mathrm{C}) \mathrm{x}^{3}+(\mathrm{B}+\mathrm{D}) \mathrm{x}^{2}+\left(-\mathrm{Ab}^{2}-\mathrm{Ca}^{2}\right) \mathrm{x}+\left(-\mathrm{b}^{2} \mathrm{~B}\right.$
$\left.-\mathrm{a}^{2} \mathrm{D}\right) \ldots \ldots$ (ii)
By equating similar terms, we get
$A+C=0 \Rightarrow A=-C \ldots \ldots \ldots \ldots$ (iii)
$B+D=0 \Rightarrow B=-D \ldots \ldots \ldots \ldots$ (iv)
$-\mathrm{Ab}^{2}-\mathrm{Ca}^{2}=1$
$\Rightarrow-(-C) b^{2}-C a^{2}=1($ from equation(iii))
$\Rightarrow \quad C=\frac{1}{b^{2}-a^{2}} \ldots \ldots \ldots \ldots \ldots(v)$
$-b^{2} B-a^{2} D=0$
$\Rightarrow-b^{2}(-D)-a^{2} D=0$
$\Rightarrow D=0$
So equation(iv) becomes $B=0$
So equation (iii) becomes, $A=-\frac{1}{b^{2}-a^{2}}$
We put the values of $A, B, C$, and $D$ values back into our partial fractions in equation (i) and replace this as the integrand. We get
$\int \frac{x}{\left(x^{2}-a^{2}\right)\left(x^{2}-b^{2}\right)} d x$
$\Rightarrow \int\left[\frac{A x+B}{\left(x^{2}-a^{2}\right)}+\frac{C x+D}{\left(x^{2}-b^{2}\right)}\right] d x$
$\Rightarrow \int\left[\frac{\left(-\frac{1}{b^{2}-a^{2}}\right) x+0}{\left(x^{2}-a^{2}\right)}+\frac{\left(\frac{1}{b^{2}-a^{2}}\right) x+0}{\left(x^{2}-b^{2}\right)}\right] d x$
Split up the integral,
$\Rightarrow-\frac{1}{b^{2}-a^{2}} \int \frac{1}{\left(x^{2}-a^{2}\right)} d x+\frac{1}{b^{2}-a^{2}} \int \frac{1}{\left(x^{2}-b^{2}\right)} d x$
Let substitute
$u=x^{2}-a^{2} \Rightarrow d u=2 d x$
$v=x^{2}-b^{2} \Rightarrow d v=2 d x$, so the above equation becomes,
$\Rightarrow-\frac{1}{b^{2}-a^{2}} \int \frac{\frac{1}{u} d u}{2}+\frac{1}{b^{2}-a^{2}} \int \frac{\frac{1}{v} d v}{2}$
$\Rightarrow-\frac{1}{2\left(b^{2}-a^{2}\right)} \int \frac{1}{u} d u+\frac{1}{2\left(b^{2}-a^{2}\right)} \int \frac{1}{v} d v$
On integrating we get
$\Rightarrow-\frac{1}{2\left(b^{2}-a^{2}\right)} \log |u|+\frac{1}{2\left(b^{2}-a^{2}\right)} \log |v|+C$
Substituting back, we get
$\Rightarrow \frac{1}{2\left(b^{2}-a^{2}\right)}\left[\log \left|x^{2}-b^{2}\right|-\log \left|x^{2}-a^{2}\right|\right]+C$
Applying the logarithm rule we get
$\Rightarrow \frac{1}{2\left(b^{2}-a^{2}\right)}\left[\log \left|\frac{x^{2}-b^{2}}{x^{2}-a^{2}}\right|\right]+C$
Note: the absolute value signs account for the domain of the natural log function $(x>0)$.
Hence,
$\int \frac{x}{\left(x^{2}-a^{2}\right)\left(x^{2}-b^{2}\right)} d x=\frac{1}{2\left(b^{2}-a^{2}\right)}\left[\log \left|\frac{x^{2}-b^{2}}{x^{2}-a^{2}}\right|\right]+C$
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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,