Evaluate the following integrals:
$\int \frac{\log (x+2)}{(x+2)^{2}} d x$
Let I $=\int \frac{\log (x+2)}{(x+2)^{2}} \mathrm{dx}$
$\frac{1}{x+2}=t$
$\frac{-1}{(x+2)^{2}} d x=d t$
$I=-\int \log \left(\frac{1}{t}\right) d t$
Using integration by parts,
$=-\int \log \mathrm{t}^{-1} \mathrm{dt}$
$=-\int 1 \times \log \mathrm{t}^{-1} \mathrm{dt}$
We know that, $\frac{d}{d t} \log t=\frac{1}{t}$ and $\int d t=t$
$\mathrm{I}=\log \mathrm{t} \int \mathrm{dt}-\int\left(\frac{\mathrm{d}}{\mathrm{dt}} \log \mathrm{t} \int \mathrm{dt}\right) \mathrm{dt}$
$=\log \mathrm{t} \int \mathrm{dt}-\int\left(\frac{1}{\mathrm{t}} \int \mathrm{dt}\right) \mathrm{dt}$
$=\mathrm{t} \log \mathrm{t}-\int \frac{1}{\mathrm{t}} \times \mathrm{t} \mathrm{dt}$
$=\operatorname{tlog} \mathrm{t}-\mathrm{t}+\mathrm{c}$
Replace the value of $t$,
$=\frac{1}{x+2}\left(\log (x+2)^{-1}-1\right)+c$
$=-\frac{1}{x+2}-\frac{\log (x+2)}{x+2}+c$
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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,