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