Evaluate the following integral:
$\int \frac{(x-1)^{2}}{x^{4}+x^{2}+1} d x$
re-writing the given equation as
$\int \frac{x^{2}-2 x+1}{x^{4}+x^{2}+1} d x$
$\int \frac{1-\frac{2}{x}+\frac{1}{x^{2}}}{x^{2}+1+\frac{1}{x^{2}}} d x$
$\int \frac{1+\frac{1}{x^{2}}}{\left(x-\frac{1}{x}\right)^{2}+3} d x-\int \frac{2 x}{x^{4}+x^{2}+1} d x$
Substituting $\mathrm{tas} \mathrm{x}-\frac{1}{\mathrm{x}}$ and $\mathrm{z}$ as $\mathrm{x}^{2}$
$\left(1+\frac{1}{x^{2}}\right) d x=d t$ and $2 x d x=d z$
$\int \frac{d t}{(t)^{2}+3}-\frac{3}{2} \int \frac{d z}{z^{2}+z+1}$
$\int \frac{d t}{(t)^{2}+3}-\int \frac{d z}{\left(z+\frac{1}{2}\right)^{2}+\frac{3}{4}}$
Using identity $\int \frac{1}{x^{2}+1} d x=\arctan (x)$
$\frac{1}{\sqrt{3}} \arctan \left(\frac{t}{\sqrt{3}}\right)-\frac{2}{\sqrt{3}} \arctan \left(\frac{2 z+1}{\sqrt{3}}\right)+c$
Substituting $\mathrm{t}$ as $\mathrm{x}-\frac{1}{\mathrm{x}}$ and $\mathrm{z}$ as $\mathrm{x}^{2}$
$\frac{1}{\sqrt{3}} \arctan \left(\frac{x-\frac{1}{x}}{\sqrt{3}}\right)-\frac{2}{\sqrt{3}} \arctan \left(\frac{2 x^{2}+1}{\sqrt{3}}\right)+c$
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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,