Evaluate $\int \sqrt{\frac{a+x}{x}} d x$
to solve this integral we have to apply substitution method for which we are going to put $x=a \cdot \tan ^{2} k$
This means $\mathrm{dx}=2$.a.tank.sec ${ }^{2} k \cdot d k$, then I will be,
$I=\int \sqrt{\frac{\operatorname{asec}^{2} k}{a \tan ^{2} k}} \cdot 2 a \cdot \tan k \cdot \sec ^{2} k \cdot d k=2 a \cdot \operatorname{cosec} k \cdot \tan k \cdot \sec ^{2} k \cdot d k$
In this above integral let $\tan k=t$ then $\sec ^{2} k d k=d t$, put in above equation-
$I=2 a \int \sqrt{\left(t^{2}+1\right)} \cdot d t$
Apply the formula of sqrt $\left(x^{2}+a^{2}\right)=x / 2 \cdot \operatorname{sqrt}\left(a^{2}+x^{2}\right)+a^{2} / 2 \ln
$I=2 a\left[\frac{t}{2} \cdot \sqrt{1+t^{2}}+\frac{1}{2} \cdot \ln \left|t+\sqrt{1+t^{2}}\right|\right]$\left|x+\operatorname{sqrt}\left(a^{2}+x^{2}\right)\right|$
Now put the value of $t$ in above integral $t=\operatorname{tank}$, then finally integral will be-
$I=2 a\left[\frac{\tan k}{2} \cdot \sqrt{1+\tan ^{2} k}+\frac{1}{2} \cdot \ln \mid \tan k+\sqrt{1+\tan ^{2} k}\right]$
Now put the value of $k$ in terms of $x$ that is $\tan ^{2} k=x / a$ in above integral -
$I=2 a\left[\frac{1}{2} \sqrt{\frac{x}{a}} \cdot \sqrt{1+\frac{x}{a}}+\frac{1}{2} \cdot \ln \mid \frac{1}{2} \sqrt{\frac{x}{a}}+\sqrt{1+\frac{x}{a}}\right]$
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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,