Evaluate $\int\left(1+x^{2}\right) \cos 2 x d x$
$y=\int \cos 2 x+x^{2} \cos 2 x d x$
$A=\int \cos 2 x d x$
$A=\frac{\sin 2 x}{2}+c_{1}$
$B=\int x^{2} \cos 2 x d x$
Use the method of integration by parts
$B=x^{2} \int \cos 2 x d x-\int \frac{d}{d x}\left(x^{2}\right)\left(\int \cos 2 x d x\right) d x$
$B=x^{2} \frac{\sin 2 x}{2}-\int x \sin 2 x d x$
$B=x^{2} \frac{\sin 2 x}{2}-\left(x \int \sin 2 x d x-\int \frac{d}{d x}(x)\left(\int \sin 2 x d x\right)\right.$
$B=x^{2} \frac{\sin 2 x}{2}+x \frac{\cos 2 x}{2}-\frac{\sin 2 x}{4}+c_{2}$
The final answer is $\mathrm{y}=\mathrm{A}+\mathrm{B}$
$y=\frac{\sin 2 x}{2}+x^{2} \frac{\sin 2 x}{2}+x \frac{\cos 2 x}{2}-\frac{\sin 2 x}{4}+c$
$y=\frac{\left(1+x^{2}\right)}{2} \sin 2 x+\frac{x}{2} \cos 2 x-\frac{1}{4} \sin 2 x+c$
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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,