Evaluate the following integrals:
$\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x$
Let $I=\int \sin ^{-1} \sqrt{\frac{x}{a+x}} d x$
Let $\mathrm{x}=\mathrm{a} \tan ^{2} \theta$
$\mathrm{dx}=2 \mathrm{a} \tan ^{2} \theta \sec ^{2} \theta$
$I=\int\left(\sin ^{-1} \sqrt{\frac{\operatorname{atan}^{2} \theta}{a+\operatorname{atan}^{2} \theta}}\right) 2 a \tan ^{2} \theta \sec ^{2} \theta d \theta$
$=\int \sin ^{-1}(\sin \theta) 2 a \tan ^{2} \theta \sec ^{2} \theta d \theta$
$=\int 2 \theta a \tan ^{2} \theta \sec ^{2} \theta d \theta$
$=2 a \int \theta \tan ^{2} \theta \sec ^{2} \theta d \theta$
Using integration by parts,
$=2 \mathrm{a}\left(\theta \int \tan ^{2} \theta \sec ^{2} \theta \mathrm{d} \theta-\int 1 \int \tan ^{2} \theta \sec ^{2} \theta \mathrm{d} \theta\right)$
$=2 \mathrm{a}\left[\theta \frac{\tan ^{2} \theta}{2}-\int \frac{\tan ^{2} \theta}{2} \mathrm{~d} \theta\right]$
$=\mathrm{a} \theta \tan ^{2} \theta-\frac{2 \mathrm{a}}{2} \int\left(\sec ^{2} \theta-1\right) \mathrm{d} \theta$
$=a \theta \tan ^{2} \theta-a \tan \theta+a \theta+c$
$=a\left(\tan ^{-1} \sqrt{\frac{x}{a}}\right) \frac{x}{a}-a \sqrt{\frac{x}{a}}+a \tan ^{-1} \sqrt{\frac{x}{a}}+c$
$=x \tan ^{-1} \sqrt{\frac{x}{a}}-\sqrt{a x}+a \tan ^{-1} \sqrt{\frac{x}{a}}+c$
- 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,