Evaluate the following integrals:
$\int e^{x}\left(\frac{1+\sin x}{1+\cos x}\right) d x$
Let I $=\int e^{x}\left(\frac{1+\sin x}{1+\cos x}\right) d x$
We know that, $\sin ^{2} x+\cos ^{2} x=1$ and $\sin x=2 \sin \frac{x}{2} \cos \frac{x}{2}$
$=e^{x}\left(\frac{\sin ^{2} \frac{x}{2}+\cos ^{2} \frac{x}{2}+2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \cos ^{2} \frac{x}{2}}\right)$
$=\frac{e^{x}\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2}}{2 \cos ^{2} \frac{x}{2}}$
$=\frac{1}{2} e^{x}\left(\frac{\sin \frac{x}{2}+\cos \frac{x}{2}}{2 \cos \frac{x}{2}}\right)^{2}$
$=\frac{1}{2} e^{x}\left[\tan \frac{x}{2}+1\right]^{2}$
$=\frac{1}{2} e^{x}\left[1+\tan \frac{x}{2}\right]^{2}$
$=\frac{1}{2} e^{x}\left[1+\tan ^{2} \frac{x}{2}+2 \tan \frac{x}{2}\right]$
$=\frac{1}{2} e^{x}\left[\sec ^{2} \frac{x}{2}+2 \tan \frac{x}{2}\right]$
$=e^{x}\left[\frac{1}{2} \sec ^{2} \frac{x}{2}+\tan \frac{x}{2}\right] \ldots \ldots(1)$
Let $\tan \frac{x}{2}=f(x)$
$f^{\prime}(x)=\frac{1}{2} \sec ^{2} \frac{x}{2}$
We know that,
$\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x=e^{x} f(x)+c$
From equation(1), we obtain
$\int e^{x}\left(\frac{1+\sin x}{1+\cos x}\right) d x=e^{x} \tan \frac{x}{2}+c$
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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,