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