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