Question:
If $\int \frac{\mathrm{x}+1}{\sqrt{2 \mathrm{x}-1}} \mathrm{dx}=\mathrm{f}(\mathrm{x}) \sqrt{2 \mathrm{x}-1}+\mathrm{C}$, where $C$ is a
constant of integration, then $\mathrm{f}(\mathrm{x})$ is equal to :-
$\frac{1}{3}(x+4)$
$\frac{1}{3}(x+1)$
$\frac{2}{3}(x+2)$
$\frac{2}{3}(x-4)$
Correct Option: 1
Solution:
$\sqrt{2 \mathrm{x}-1}=\mathrm{t} \Rightarrow 2 \mathrm{x}-1=\mathrm{t}^{2} \Rightarrow 2 \mathrm{dx}=2 \mathrm{t} \cdot \mathrm{dt}$
$\int \frac{x+1}{\sqrt{2 x-1}} d x=\int \frac{\frac{t^{2}+1}{2}+1}{t} t d t=\int \frac{t^{2}+3}{2} d t$
$=\frac{1}{2}\left(\frac{t^{3}}{3}+3 t\right)=\frac{t}{6}\left(t^{2}+9\right)+c$
$=\sqrt{2 x-1}\left(\frac{2 x-1+9}{6}\right)+c=\sqrt{2 x-1}\left(\frac{x+4}{3}\right)+c$
$\Rightarrow f(x)=\frac{x+4}{3}$
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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,
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