Evaluate the following integral:
$\int \frac{1}{\sin x(3+2 \cos x)} d x$
Let, $\mathrm{I}=\int \frac{1}{\sin \mathrm{x}(3+2 \cos \mathrm{x})} \mathrm{dx}$
Multiplying and dividing by $\sin x$
$\therefore \mathrm{I}=\int \frac{\sin \mathrm{x}}{\sin ^{2} \mathrm{x}(3+2 \cos \mathrm{x})} \mathrm{dx}$
$\therefore \mathrm{I}=\int \frac{\sin \mathrm{x}}{\left(1-\cos ^{2} \mathrm{x}\right)(3+2 \cos \mathrm{x})} \mathrm{dx}$
Let $\cos x=t,-\sin x d x=d t$
So, $I=\int \frac{d t}{\left(t^{2}-1\right)(3+2 t)}$
Now, let $\frac{1}{\left(t^{2}-1\right)(3+2 t)}=\frac{A}{t-1}+\frac{B}{t+1}+\frac{C}{3+2 t}$
$\Rightarrow 1=A(t+1)(3+2 t)+B(t-1)(3+2 t)+C\left(t^{2}-1\right)$
For, $\mathrm{t}=1, \mathrm{~A}=\frac{1}{10}$
For, $t=-1, B=-\frac{1}{2}$
For, $t=-\frac{3}{2}, C=\frac{4}{5}$
$\therefore \mathrm{I}=\frac{1}{10} \int \frac{\mathrm{dt}}{\mathrm{t}-1}-\frac{1}{2} \int \frac{\mathrm{dt}}{\mathrm{t}+1}+\frac{4}{5} \int \frac{\mathrm{dt}}{3+2 \mathrm{t}}$
$\Rightarrow I=\frac{1}{10} \log |t-1|-\frac{1}{2} \log |t+1|+\frac{2}{5} \log |3+2 t|+c$
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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,