Question:
Evaluate the following integrals:
$\int x \sin ^{3} x d x$
Solution:
Let $\mathrm{I}=\int \mathrm{x} \sin ^{3} \mathrm{x} \mathrm{dx}$
We know that, $\sin ^{3} x=\frac{3 \sin x-\sin 3 x}{4}$
$=\int x\left(\frac{3 \sin x-\sin 3 x}{4}\right) d x$
$=\frac{1}{4} \int x(3 \sin x-\sin 3 x) d x$
Using integration by parts,
$I=\frac{1}{4}\left[x \int(3 \sin x-\sin 3 x) d x-\int 1 \int(3 \sin x-\sin 3 x) d x\right]$
$=\frac{1}{4}\left[x\left(-3 \cos x+\frac{\cos 3 x}{3}\right)-\int\left(-3 \cos x+\frac{\cos 3 x}{3}\right) d x\right]$
$=\frac{1}{4}\left[-3 x \cos x+\frac{x \cos 3 x}{3}+3 \sin x-\frac{\sin 3 x}{9}\right]+c$
$I=\frac{1}{36}[3 x \cos 3 x-27 x \cos x+27 \sin x-\sin 3 x]+c$
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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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