Question:
Evaluate the following integral:
$\int \frac{1}{(x-1) \sqrt{2 x+3}} d x$
Solution:
assume $2 x+3=t^{2}$
$\mathrm{d} \mathrm{x}=\mathrm{tdt}$
$\int \frac{d t}{\frac{t^{2}-3}{2}-1}$
$\int \frac{2 d t}{\left(t^{2}-5\right)}$
Using identity $\int \frac{\mathrm{dz}}{(\mathrm{z})^{2}-1}=\frac{1}{2} \log \left|\frac{z-1}{z+1}\right|+c$
$\frac{1}{\sqrt{5}} \log \left|\frac{t-\sqrt{5}}{t+\sqrt{5}}\right|+c$
$\frac{1}{\sqrt{5}} \log \left|\frac{\sqrt{(2 x+3)}-\sqrt{5}}{\sqrt{2 x+3}+\sqrt{5}}\right|+c$
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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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