Evaluate the integral
$\int \frac{4 \sin x+5 \cos x}{5 \sin x+4 \cos x} d x$
Ideas required to solve the problems:
* Integration by substitution: A change in the variable of integration often reduces an integral to one of the fundamental integration. If derivative of a function is present in an integration or if chances of its presence after few modification is possible then we apply integration by substitution method.
* Knowledge of integration of fundamental functions like sin, cos, polynomial, log etc and formula for some special functions.
Let, $I=\int \frac{4 \sin x+5 \cos x}{5 \sin x+4 \cos x} d x$
To solve such integrals involving trigonometric terms in numerator and denominators. We use the basic substitution method and to apply this simply we follow the undermentioned procedure-
If I has the form $\int \frac{a \sin x+b \cos x+c}{d \sin x+e \cos x+f} d x$
Then substitute numerator as -
$a \sin x+b \cos x+c=A \frac{d}{d x}(d \sin x+e \cos x+f)+B(d \sin x+e \cos x+c)+C$
Where A, B and C are constants
We have, $I=\int \frac{4 \sin x+5 \cos x}{5 \sin x+4 \cos x} d x$
As I matches with the form described above, So we will take the steps as described.
$\therefore 4 \sin x+5 \cos x=A \frac{d}{d x}(5 \sin x+4 \cos x)+B(4 \cos x+5 \sin x)+C$
$\Rightarrow 4 \sin x+5 \cos x=A(5 \cos x-4 \sin x)+B(4 \cos x+5 \sin x)+C \quad\left\{\because \frac{d}{d x} \cos x=-\sin x\right\}$
$\Rightarrow 4 \sin x+5 \cos x=\sin x(5 B-4 A)+\cos x(5 A+4 B)+C$
Comparing both sides we have:
$C=0$
$5 B-4 A=4$
$4 B+5 A=5$
On solving for $A, B$ and $C$ we have:
$A=9 / 41, B=40 / 41$ and $C=0$
Thus I can be expressed as:
$I=\int \frac{\frac{9}{41}(5 \cos x-4 \sin x)+\frac{40}{41}(4 \cos x+5 \sin x)}{4 \cos x+5 \sin x} d x$
$I=\int \frac{\frac{9}{41}(5 \cos x-4 \sin x)}{4 \cos x+5 \sin x} d x+\int \frac{\frac{40}{41}(4 \cos x+5 \sin x)}{4 \cos x+5 \sin x} d x$
$\therefore$ Let $\mathrm{I}_{1}=\frac{9}{41} \int \frac{(5 \cos \mathrm{x}-4 \sin \mathrm{x})}{4 \cos \mathrm{x}+5 \sin \mathrm{x}}$ and $\mathrm{I}_{2}=\frac{40}{41} \int \frac{(4 \cos \mathrm{x}+5 \sin \mathrm{x})}{4 \cos \mathrm{x}+5 \sin \mathrm{x}} \mathrm{dx}$
$\Rightarrow \mathrm{I}=\mathrm{I}_{1}+\mathrm{I}_{2} \ldots$ equation 1
$I_{1}=\frac{9}{41} \int \frac{(5 \cos x-4 \sin x)}{4 \cos x+5 \sin x}$
Let, $4 \cos x+5 \sin x=u$
$\Rightarrow(-4 \sin x+5 \cos x) d x=d u$
So, $I_{1}$ reduces to:
$I_{1}=\frac{9}{41} \int \frac{\mathrm{du}}{\mathrm{u}}=\frac{9}{41} \log |\mathrm{ul}|+C_{1}$
$\therefore \mathrm{I}_{1}=\frac{9}{41} \log |4 \cos \mathrm{x}+5 \sin \mathrm{x}|+\mathrm{C}_{1} \ldots \ldots$ equation 2
As, $I_{2}=\frac{40}{41} \int \frac{(4 \cos x+5 \sin x)}{4 \cos x+5 \sin x} d x$
$\Rightarrow I_{2}=\frac{40}{41} \int d x=\frac{40 x}{41}+C_{2} \ldots . .$ equation 3
From equation 1,2 and 3 we have:
$I=\frac{9}{41} \log |4 \cos x+5 \sin x|+C_{1}+\frac{40 x}{41}+C_{2}$
$\therefore I=\frac{9}{41} \log |4 \cos x+5 \sin x|+\frac{40 x}{41}+C$
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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,