Evaluate the following integrals:
$\int \frac{x^{2} \sin ^{-1} x}{\left(1-x^{2}\right)^{3 / 2}} d x$
Let I $=\int \frac{x^{2} \sin ^{-1} x}{\left(1-x^{2}\right)^{3 / 2}} d x$
$\sin ^{-1} x=t$
$\frac{1}{\sqrt{1-x^{2}}} d x=d t$
$I=\int \frac{\sin ^{2} t \times t d t}{1-\sin ^{2} t}$
$=\int \frac{\operatorname{tsin}^{2} t}{\cos ^{2} t} d t$
$=\int t \tan ^{2} t d t$
$=\int t\left(\sec ^{2} t-1\right) d t$
Using integration by parts,
$=\int \operatorname{tsec}^{2} t d t-\int t d t$
$=t \int \sec ^{2} t d t-\int \frac{d}{d t} t \int \sec ^{2} t d t-\frac{t^{2}}{2}$
We know that, $\int \sec ^{2} t d t=\tan t$
$=\operatorname{ttan} t-\int \tan t d t-\frac{t^{2}}{2}$
$=\operatorname{ttan} t-\log |\sec t|-\frac{t^{2}}{2}+c$
$I=\frac{x}{\sqrt{1-x^{2}}} \sin ^{-1} x+\log \left|1-x^{2}\right|-\frac{1}{2}\left(\sin ^{-1} x\right)^{2}+c$
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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,