Evaluate the following integrals:
$\int \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}} d x$
Let $I=\int \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}} d x$
We are splitting this in to two functions
First we find the integral of:
$\int \frac{\mathrm{x}}{\sqrt{1-\mathrm{x}^{2}}} \mathrm{dx}$
Put $1-x^{2}=t$
$-2 x d x=d t$
$\int \frac{\mathrm{x}}{\sqrt{1-\mathrm{x}^{2}}} \mathrm{dx}=-\frac{1}{2} \int \frac{\mathrm{dt}}{\sqrt{\mathrm{t}}}=-\sqrt{\mathrm{t}}=-\sqrt{1-\mathrm{x}^{2}}$
$I=\int \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}} d x$
Using integration by parts,
$=\left(\sin ^{-1} x\right) \times-\sqrt{1-x^{2}}-\int \frac{1}{\sqrt{1-x^{2}}}\left(-\sqrt{1-x^{2}}\right) d x$
$=\left(\sin ^{-1} x\right) \times-\sqrt{1-x^{2}}-\int d x$
$=\left(\sin ^{-1} x\right) x-\sqrt{1-x^{2}}+x+c$
$=x-\sqrt{1-x^{2}}\left(\sin ^{-1} x\right)+c$
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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,