Question:
Evaluate the following integrals:
$\int x^{2} \cos x d x$
Solution:
Let I $=\int x^{2} \cos x d x$
Using integration by parts,
$=x^{2} \int \cos x d x-\int \frac{d}{d x} x^{2} \int \cos x d x$
We know that, $\int \cos x d x=\sin x$ and $\frac{d}{d x} x^{n}=n x^{n-1}$
$=x^{2} \sin x-\int 2 x \sin x d x$
$=x^{2} \sin x-2 \int x \sin x d x$
We know that, $\int \sin x d x=-\cos x$
$=x^{2} \sin x-2\left(x \int \sin x d x-\int \frac{d}{d x} x \int \sin x d x\right)$
$=x^{2} \sin x-2\left(-x \cos x+\int \cos x d x\right)$
$=x^{2} \sin x-2(-x \cos x+\sin x)+c$
$=x^{2} \sin x+2 x \cos x-2 \sin x+c$
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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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