Evaluate the following integrals:
$\int x^{2} \sin ^{-1} x d x$
Let $I=\int x^{2} \sin ^{-1} x d x$
Using integration by parts,
$I=\sin ^{-1} x \int x^{2} d x-\int \frac{d}{d x} \sin ^{-1} x \int x^{2} d x$
$=\frac{x^{3}}{3} \sin ^{-1} x-\int \frac{x^{3}}{3 \sqrt{1-x^{2}}} d x$
$I=\frac{x^{3}}{3} \sin ^{-1} x-\int I_{1}+C$ .......(i)
$\mathrm{I}_{1}=-\int \frac{\mathrm{x}^{3}}{3 \sqrt{1-\mathrm{x}^{2}}} \mathrm{dx}$
Let $1-x^{2}=t^{2}$
$-2 x d x=2 t d t$
$-x d x=t d t$
$\mathrm{I}_{1}=-\int \frac{\left(1-\mathrm{t}^{2}\right) \mathrm{tdt}}{\mathrm{t}}$
$\mathrm{I}_{1}=\int\left(\mathrm{t}^{2}-1\right) \mathrm{dt}$
$=\frac{\mathrm{t}^{3}}{3}-\mathrm{t}+\mathrm{c}_{2}$
$=\frac{\left(1-x^{2}\right)^{\frac{3}{2}}}{3}-\left(1-x^{2}\right)^{\frac{1}{2}}+c_{2}$
$=\frac{x^{3}}{3} \sin ^{-1} x-\frac{\left(1-x^{2}\right)^{\frac{3}{2}}}{9}+\frac{1}{3}\left(1-x^{2}\right)^{\frac{1}{2}}+c$
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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,