Evaluate the following integral:
$\int \frac{1}{\left(x^{2}+1\right)\left(x^{2}+2\right)} d x$
Let, $I=\int \frac{1}{\left(x^{2}+1\right)\left(x^{2}+2\right)} d x$
Let, $x^{2}=y$
Then, $\frac{1}{(\mathrm{y}+1)(\mathrm{y}+2)}=\frac{\mathrm{A}}{\mathrm{y}+1}+\frac{\mathrm{B}}{\mathrm{y}+2}$
$\Rightarrow 1=A(y+2)+B(y+1)$
$\Rightarrow 1=(A+B) y+2 A+B$
On equating similar terms, we get,
$A+B=0$, and $2 A+B=1$
We get, $A=1, B=-1$
$\therefore \mathrm{I}=\int \frac{\mathrm{dx}}{\mathrm{x}^{2}+1}-\int \frac{\mathrm{dx}}{\mathrm{x}^{2}+2}$
$\Rightarrow \mathrm{I}=\tan ^{-1} \mathrm{x}-\frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{\mathrm{x}}{\sqrt{2}}\right)+\mathrm{c}$
So, $\int \frac{1}{\left(x^{2}+1\right)\left(x^{2}+2\right)} d x=\tan ^{-1} x-\frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)+c$
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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,