Evaluate the following integral:
$\int \frac{1}{x\left(x^{3}+8\right)} d x$
Consider the integral,
$I=\int \frac{1}{x\left(x^{3}+8\right)} d x$
Rewriting the above integral, we have
$I=\int \frac{x^{2}}{x^{3}\left(x^{3}+8\right)} d x$
$I=\frac{1}{3} \int \frac{3 x^{2}}{x^{3}\left(x^{3}+8\right)} d x$
Substitute $x^{3}=t$
$3 x^{2} d x=d t$
$I=\frac{1}{3} \int \frac{d t}{t(t+8)}$
$\frac{1}{t(t+8)}=\frac{A}{t}+\frac{B}{t+8}$
$1=A(t+8)+B t$
Equating constants
$1=8 \mathrm{~A}$
$\mathrm{~A}=\frac{1}{8}$
Equating coefficients of $t$
$0=A+B$
$B=-\frac{1}{8}$
$I=\frac{1}{3} \int \frac{d t}{t(t+8)}$
$=\frac{1}{3} \int\left(\frac{\frac{1}{8}}{t}-\frac{\frac{1}{8}}{t+8}\right) d t$
$=\frac{1}{3} \times \frac{1}{8} \int \frac{\mathrm{dt}}{\mathrm{t}}-\frac{1}{3} \times \frac{1}{8} \int \frac{\mathrm{dt}}{\mathrm{t}+8}$
$=\frac{1}{24} \log \mathrm{t}-\frac{1}{24} \log |\mathrm{t}+8|+\mathrm{C}$
$=\frac{1}{24} \log \mathrm{x}^{3}-\frac{1}{24} \log \left|\mathrm{x}^{3}+8\right|+\mathrm{C}$
$=\frac{3}{24} \log \mathrm{x}-\frac{1}{24} \log \left|\mathrm{x}^{3}+8\right|+\mathrm{C}$
$=\frac{1}{8} \log \mathrm{x}-\frac{1}{24} \log \left|\mathrm{x}^{3}+8\right|+\mathrm{C}$
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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,