Question:
Evaluate the following integrals:
$\int \frac{1}{x^{2}\left(x^{4}+1\right)^{3 / 4}} d x$
Solution:
$I=\int \frac{1}{x^{2}\left(x^{4}+1\right)^{\frac{3}{4}}} d x$
$\Rightarrow \int \frac{1}{x^{5}\left(1+\frac{1}{x^{4}}\right)^{\frac{3}{4}}} d x$
Let $1+\frac{1}{x^{4}}=t$
$\Rightarrow-\frac{4}{x^{5}} d x=d t$
$\Rightarrow \frac{1}{x^{5}} d x=\frac{-d t}{4}$
$I=\frac{-1}{4} \int \frac{1}{t^{\frac{3}{4}}} d t$
$\Rightarrow \frac{-1}{4}\left(\frac{\mathrm{t}^{\frac{1}{4}}}{\frac{1}{4}}\right)+\mathrm{C}$
$\Rightarrow-t^{\frac{1}{4}}+c$
But $t=1+\frac{1}{x^{4}}$
$\Rightarrow-\left(1+\frac{1}{x^{4}}\right)^{\frac{1}{4}}+c$
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- 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
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- Chemical Effects of Electric Current,
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