Question:
Evaluate the following integral:
$\int \frac{3 x+5}{x^{3}-x^{2}-x+1} d x$
Solution:
$I=\int \frac{3 x+5}{x^{3}-x^{2}-x+1} d x=\int \frac{3 x+5}{(x-1)^{2}(x+1)}$
$\frac{3 x+5}{(x-1)^{2}(x+1)}=\frac{A}{x-1}+\frac{B}{(x-1)^{2}}+\frac{C}{x+1}$
$3 x+5=A(x-1)(x+1)+B(x+1)+C(x-1)^{2}$
Put $x=1$
$-3+5=4 C$
$2=4 \mathrm{C}$
$C=\frac{1}{2}$
Put $x=0$
$5=-A+B+C$
$A=\frac{1}{2}$
$\int \frac{3 x+5}{(x-1)^{2}(x+1)} d x=\frac{1}{2} \int \frac{d x}{x-1}+4 \int \frac{d x}{(x-1)^{2}}+\frac{1}{2} \int \frac{d x}{x+1}$
$=-\frac{1}{2} \ln |x-1|-\frac{4}{(x-1)}+\frac{1}{2} \ln |x+1|+C$
$=\frac{1}{2} \ln \left|\frac{x+1}{x-1}\right|-\frac{4}{(x-1)}+C$
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- 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,
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