Question:
Evaluate the following integrals:
$\int x \sin x \cos 2 x d x$
Solution:
Let $I=\int x \sin x \cos 2 x d x=\frac{1}{2} \int x \times 2 \sin x \cos 2 x d x$
Using integration by parts,
$=\frac{1}{2} \int x(\sin (x+2 x)-\sin (2 x-x)) d x$
$=\frac{1}{2} \int x(\sin 3 x-\sin x) d x$
Using integration by parts,
$=\frac{1}{2}\left(x \int(\sin 3 x-\sin x) d x-\int \frac{d}{d x} x \int(\sin 3 x-\sin x) d x\right) d x$
$=\frac{1}{2}\left[x\left(\frac{-\cos 3 x}{3}+\cos x\right)-\int-\left(\frac{\cos 3 x}{3}+\cos x\right) d x\right)$
$I=\frac{1}{2}\left[-x \frac{\cos 3 x}{3}+x \cos x+\frac{1}{9} \sin 3 x-\sin x\right]+c$
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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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