Evaluate the following integrals:
$\int \sin ^{-1}\left(3 x-4 x^{3}\right) d x$
Let $\int \sin ^{-1}\left(3 x-4 x^{3}\right) d x$
$x=\sin \theta$
$\mathrm{dx}=\cos \theta \mathrm{d} \theta$
$=\int \sin ^{-1}\left(3 \sin \theta-4 \sin ^{3} \theta\right) \cos \theta d \theta$
We know that $3 \sin \theta-4 \sin ^{3} \theta=\sin 3 \theta$
$=\int \sin ^{-1}(\sin 3 \theta) \cos \theta d \theta$
We know that, $\int \sin ^{-1}(\sin 3 \theta)=3 \theta$
$=\int 3 \theta \cos \theta d \theta$
$=3 \int \theta \cos \theta d \theta$
Using integration by parts,
$=3\left(\theta \int \cos \theta d \theta-\int \frac{d}{d \theta} \theta \int \cos \theta d \theta\right)$
$=3\left(\theta \times \sin \theta-\int \sin \theta d \theta\right)$
$=3(\theta \sin \theta+\cos \theta)+c$
$I=3\left[x \sin ^{-1} x+\sqrt{1-x^{2}}\right]+c$
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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,