$\int_{0}^{\frac{2}{3}} \frac{d x}{4+9 x^{2}}$ equals
A. $\frac{\pi}{6}$
B. $\frac{\pi}{12}$
C. $\frac{\pi}{24}$
D. $\frac{\pi}{4}$
$\int \frac{d x}{4+9 x^{2}}=\int \frac{d x}{(2)^{2}+(3 x)^{2}}$
Put $3 x=t \Rightarrow 3 d x=d t$
$\therefore \int \frac{d x}{(2)^{2}+(3 x)^{2}}=\frac{1}{3} \int \frac{d t}{(2)^{2}+t^{2}}$
$=\frac{1}{3}\left[\frac{1}{2} \tan ^{-1} \frac{t}{2}\right]$
$=\frac{1}{6} \tan ^{-1}\left(\frac{3 x}{2}\right)$
$=\mathrm{F}(x)$
By second fundamental theorem of calculus, we obtain
$\int_{0}^{2} \frac{d x}{4+9 x^{2}}=\mathrm{F}\left(\frac{2}{3}\right)-\mathrm{F}(0)$
$=\frac{1}{6} \tan ^{-1}\left(\frac{3}{2} \cdot \frac{2}{3}\right)-\frac{1}{6} \tan ^{-1} 0$
$=\frac{1}{6} \tan ^{-1} 1-0$
$=\frac{1}{6} \times \frac{\pi}{4}$
$=\frac{\pi}{24}$
Hence, the correct answer is 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,