$\int \sqrt{\frac{a+x}{a-x}}$
Let, $I=\int \sqrt{\frac{a+x}{a-x}} d x$
$=\int \sqrt{\frac{a+x}{a-x} \times \frac{a+x}{a+x}} d x=\int \frac{a+x}{\sqrt{(a-x)(a+x)}} d x$
$=\int \frac{a+x}{\sqrt{a^{2}-x^{2}}} d x=\int \frac{a}{\sqrt{a^{2}-x^{2}}} d x+\int \frac{x}{\sqrt{a^{2}-x^{2}}} d x$
Let's consider, $\mathrm{I}=\mathrm{I}_{1}+\mathrm{I}_{2}$
Now, $\mathrm{I}_{1}=\int \frac{a}{\sqrt{a^{2}-x^{2}}} d x=a \cdot \sin ^{-1} \frac{x}{a}+\mathrm{C}_{1}$
And, $\mathrm{I}_{2}=\int \frac{x}{\sqrt{a^{2}-x^{2}}} d x$
Putting, $a^{2}-x^{2}=t \Rightarrow-2 x d x=d t$
$x d x=\frac{d t}{-2}$
Hence, $I_{2}=-\frac{1}{2} \int \frac{d t}{\sqrt{t}}=-\frac{1}{2} \times 2 \sqrt{t}=-\sqrt{a^{2}-x^{2}}+C_{2}$
As, $I=I_{1}+I_{2}$
$=a \sin ^{-1} \frac{x}{a}+C_{1}-\sqrt{a^{2}-x^{2}}+C_{2}$
$\therefore \mathrm{I}=a \sin ^{-1} \frac{x}{a}-\sqrt{a^{2}-x^{2}}+\left(\mathrm{C}_{1}+\mathrm{C}_{2}\right)$
Therefore, $\mathrm{I}=a \sin ^{-1} \frac{x}{a}-\sqrt{a^{2}-x^{2}}+\mathrm{C}$ $\left[\mathrm{C}=\mathrm{C}_{1}+\mathrm{C}_{2}\right]$
- 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,
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,