Question:
Evaluate the following integrals:
$\int \frac{x^{3} \sin ^{-1} x^{2}}{\sqrt{1-x^{4}}} d x$
Solution:
Let $I=\int \frac{x^{3} \sin ^{-1} x^{2}}{\sqrt{1-x^{4}}} d x$
$\sin ^{-1} x^{2}=t$
$\frac{1}{\sqrt{1-x^{4}}} 2 x d x=d t$
$I=\int \frac{x^{2} \sin ^{-1} x^{2}}{\sqrt{1-x^{4}}} x d x$
$=\int(\sin t) t \frac{d t}{2}$
Using integration by parts,
$=\frac{1}{2}\left[\mathrm{t} \int \sin t \mathrm{~d} t-\int \frac{\mathrm{d}}{\mathrm{dt}} \mathrm{t} \int \sin t \mathrm{dt}\right]$
$=\frac{1}{2}\left[-\mathrm{t} \cos t-\int-\mathrm{costdt}\right]$
$=\frac{1}{2}[-t \cos t+\sin t]+c$
$=\frac{1}{2}\left[x^{2}-\sqrt{1-x^{4}} \sin ^{-1} x^{2}\right]+c$
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- 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,
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