$\int_{0}^{\frac{\pi}{4}} \log (1+\tan x) d x$
Let $I=\int_{0}^{\frac{\pi}{4}} \log (1+\tan x) d x$ ...(1)
$\therefore I=\int_{0}^{\frac{\pi}{4}} \log \left[1+\tan \left(\frac{\pi}{4}-x\right)\right] d x$ $\left(\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x\right)$
$\Rightarrow I=\int_{0}^{\frac{\pi}{4}} \log \left\{1+\frac{\tan \frac{\pi}{4}-\tan x}{1+\tan \frac{\pi}{4} \tan x}\right\} d x$
$\Rightarrow I=\int_{0}^{\frac{\pi}{4}} \log \left\{1+\frac{1-\tan x}{1+\tan }\right\} d x$
$\Rightarrow I=\int_{0}^{\frac{\pi}{4}} \log \frac{2}{(1+\tan x)} d x$
$\Rightarrow I=\int_{0}^{\frac{\pi}{4}} \log 2 d x-\int_{0}^{\frac{\pi}{4}} \log (1+\tan x) d x$
$\Rightarrow I=\int_{0}^{\frac{\pi}{4}} \log 2 d x-I$ $[$ From (1) $]$
$\Rightarrow 2 I=[x \log 2]_{0}^{\frac{\pi}{4}}$
$\Rightarrow 2 I=\frac{\pi}{4} \log 2$
$\Rightarrow I=\frac{\pi}{8} \log 2$
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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,