Evaluate the following integrals:
$\int e^{2 x} \sin (3 x+1) d x$
Let $I=\int e^{2 x} \sin (3 x+1) d x$
Now Integrating by parts choosing $\sin (3 x+1)$ as first function and $e^{2 x}$ as second function we get,
$I=\sin (3 x+1) \int e^{2 x} d x-\int\left(\frac{d}{d x} \sin (3 x+1) \int e^{2 x} d x\right) d x$
$I=\frac{e^{2 x}}{2} \sin (3 x+1)-\int \frac{3 e^{2 x}}{2} \cos (3 x+1) d x$
Now again integrating by parts by taking $\cos (3 x+1)$ as first function and $\mathrm{e}^{2 x}$ as second function we get,
$I=\frac{e^{2 x}}{2} \sin (3 x+1)-\left[\cos (3 x+1) \int \frac{3 e^{2 x}}{2} d x-\int \frac{3}{2}\left(\frac{d}{d x} \cos (3 x+1) \int e^{2 x} d x\right) d x\right.$
$I=\frac{e^{2 x}}{2} \sin (3 x+1)-\frac{3}{4} e^{2 x} \cos (3 x+1)-\frac{9}{4} \int e^{2 x} \sin (3 x+1) d x$
$\int e^{2 x} \sin (3 x+1) d x=I$
Therefore,
$I=\frac{e^{2 x}}{2} \sin (3 x+1)-\frac{3}{4} e^{2 x} \cos (3 x+1)-\frac{9}{4} 1$
$I+\frac{9}{4} I=\frac{e^{2 x}}{2} \sin (3 x+1)-\frac{3}{4} e^{2 x} \cos (3 x+1)$
$\frac{13 I}{4}=\frac{e^{2 x}}{2} \sin (3 x+1)-\frac{3}{4} e^{2 x} \cos (3 x+1)$
$I=\frac{e^{2 x}}{13}\{2 \sin (3 x+1)-3 \cos (3 x+1)\}+c$
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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,