Question:
The integral $\int \sec ^{2 / 3} x \operatorname{cosec}{ }^{4 / 3} x d x$ is equal to
(Hence $\mathrm{C}$ is a constant of integration)
$3 \tan ^{-1 / 3} x+C$
$-\frac{3}{4} \tan ^{-4 / 3} x+C$
$-3 \cot ^{-1 / 3} x+C$
$-3 \tan ^{-1 / 3} x+C$
Correct Option: , 4
Solution:
$I=\int \frac{d x}{(\sin x)^{4 / 3} \cdot(\cos x)^{2 / 3}}$
$I=\int \frac{d x}{\left(\frac{\sin x}{\cos x}\right)^{4 / 3} \cdot \cos ^{2} x}$
$\Rightarrow I=\int \frac{\sec ^{2} x}{(\tan x)^{4 / 3}} d x$
put $\tan x=t \Rightarrow \sec ^{2} x d x=d t$
$\therefore \mathrm{I}=\int \frac{\mathrm{dt}}{\mathrm{t}^{4 / 3}} \Rightarrow \mathrm{I}=\frac{-3}{\mathrm{t}^{1 / 3}}+\mathrm{c}$
$\Rightarrow I=\frac{-3}{(\tan x)^{1 / 3}}+c$
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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,
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