Evaluate the following integral:
$\int \frac{\sin 2 x}{(1+\sin x)(2+\sin x)} d x$
The denominator is factorized, so let separate the fraction through partial fraction, hence let
$\frac{\sin 2 x}{(1+\sin x)(2+\sin x)}=\frac{A}{(1+\sin x)}+\frac{B}{2+\sin x} \ldots \ldots$ (i)
$\Rightarrow \frac{\sin 2 x}{(1+\sin x)(2+\sin x)}=\frac{A(2+\sin x)+B(1+\sin x)}{(1+\sin x)(2+\sin x)}$
$\Rightarrow \sin 2 x=A(2+\sin x)+B(1+\sin x)=2 A+A \sin x+B+B \sin x$
$\Rightarrow 2 \sin x \cos x=\sin x(A+B)+(2 A+B) \ldots \ldots$ (ii)
We need to solve for $A$ and $B$.
We will equate similar terms, we get.
$2 A+B=0 \Rightarrow B=-2 A$
And $A+B=2 \cos x$
Substituting the value of $B$, we get
$A-2 A=2 \cos x \Rightarrow A=-2 \cos x$
Hence $B=-2 A=-2(-2 \cos x)$
$\Rightarrow B=4 \cos x$
We put the values of $A$ and $B$ values back into our partial fractions in equation (i) and replace this as the integrand. We get
$\int\left[\frac{\sin 2 x}{(1+\sin x)(2+\sin x)}\right] d x$
$\Rightarrow \int\left[\frac{A}{(1+\sin x)}+\frac{B}{2+\sin x}\right] d x$
$\Rightarrow \int\left[\frac{-2 \cos x}{(1+\sin x)}+\frac{4 \cos x}{2+\sin x}\right] d x$
Split up the integral,
$\Rightarrow-\int \frac{2 \cos x}{(1+\sin x)} d x+\int \frac{4 \cos x}{2+\sin x} d x$
Let substitute
$u=\sin x \Rightarrow d u=\cos x d x$
so the above equation becomes,
$\Rightarrow-2 \int \frac{1}{(1+u)} d u+4 \int \frac{1}{2+u} d u$
Now substitute
$v=1+u \Rightarrow d v=d u$
$z=2+u \Rightarrow d z=d u$
So above equation becomes,
$\Rightarrow-2 \int \frac{1}{(v)} d v+4 \int \frac{1}{z} d z$
On integrating we get
$\Rightarrow-2 \log |v|+4 \log |z|+C$
Substituting back, we get
$\Rightarrow 4 \log |2+u|-2 \log |1+u|+C$
$\Rightarrow 4 \log |2+\sin x|-2 \log |1+\sin x|+C$
Applying logarithm rule, we get
$\Rightarrow \log \left|(2+\sin x)^{4}\right|-\log \left|(1+\sin x)^{2}\right|+C$
$\Rightarrow \log \left|\frac{(2+\sin x)^{4}}{(1+\sin x)^{2}}\right|+C$
Note: the absolute value signs account for the domain of the natural log function $(x>0)$.
Hence,
$\int \frac{\sin 2 x}{(1+\sin x)(2+\sin x)} d x=\log \left|\frac{(2+\sin x)^{4}}{(1+\sin x)^{2}}\right|+C$
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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,