Question:
Evaluate the following integral:
$\int \frac{3 x-2}{(x+1)^{2}(x+3)}$
Solution:
$I=\int \frac{3 x-2}{(x+1)^{2}(x+3)} d x$
$\frac{3 x-2}{(x+1)^{2}(x+3)}=\frac{A}{x+1}+\frac{B}{(x+1)^{2}}+\frac{c}{x+3}$
$3 x-2=A(x+1)(x+3)+B(x+3)+C(x+1)^{2}$
Put $x=-1$
$-3-2=A \times 0+B \times(-1+3)+C \times 0$
$-5=2 B$
$B=-\frac{5}{2}$
Put $x=-3$
$-9-2=C \times(-2)(-2)$
$-11=4 C$
$C=-\frac{11}{4}$
Equating coefficients of constants
$-2=3 A+3 B+C$
$-2=3 A+3 \times \frac{-5}{2}-\frac{11}{4}$
$A=\frac{11}{4}$
Thus,
$I=\frac{11}{4} \int \frac{d x}{x+1}-\frac{5}{2} \int \frac{d x}{(x+1)^{2}}-\frac{11}{4} \int \frac{d x}{x+3}$
$I=\frac{11}{4} \log |x+1|-\frac{5}{2(x+1)}-\frac{11}{4} \log |x+3|+C$
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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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