Evaluate the following integrals:
$\int x^{3} \tan ^{-1} x d x$
Let $I=\int x^{3} \tan ^{-1} x d x$
Using integration by parts,
We know that,
$\frac{d}{d x} \tan ^{-1} x=\frac{1}{2\left(1+x^{2}\right)}$
$=\tan ^{-1} \mathrm{x} \int \mathrm{x}^{3} \mathrm{dx}-\int\left(\frac{1}{1+\mathrm{x}^{2}}\right) \int \mathrm{x}^{3} \mathrm{dx}$
$=\tan ^{-1} x \frac{x^{4}}{4}-\frac{1}{4} \int \frac{x^{4}}{1+x^{2}} d x$
$\frac{1}{4} \int \frac{\mathrm{x}^{4}}{1+\mathrm{x}^{2}} \mathrm{dx}=\frac{1}{4}\left[\int \frac{1}{1+\mathrm{x}^{2}} \mathrm{dx}+\left(\mathrm{x}^{2}-1\right) \mathrm{dx}\right]=\frac{1}{4}\left[\tan ^{-1} \mathrm{x}+\frac{\mathrm{x}^{3}}{3}-\mathrm{x}\right]$
$=\frac{x^{4}}{4} \tan ^{-1} x-\frac{1}{4}\left[\tan ^{-1} x+\frac{x^{3}}{3}-x\right]+c$
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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,