Evaluate $\int \sqrt{1+2 \mathrm{x}-3 \mathrm{x}^{2}} \mathrm{dx}$
Make perfect square of quadratic equation
$1+2 x-3 x^{2}=3\left[-\left(x^{2}-\frac{2}{3} x-\frac{1}{3}\right)\right]$
$=3\left[\frac{4}{9}-\left(x^{2}-2\left(\frac{1}{3}\right)(x)+\left(\frac{1}{3}\right)^{2}\right)\right]$
$=3\left[\left(\frac{2}{3}\right)^{2}-\left(x-\frac{1}{3}\right)^{2}\right]$
$y=\sqrt{3} \int\left[\left(\frac{2}{3}\right)^{2}-\left(x-\frac{1}{3}\right)^{2}\right] d x$
Using formula, $\int \sqrt{a^{2}-x^{2}} d x=\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+\frac{x}{2} \sqrt{a^{2}-x^{2}}$
$y=\sqrt{3}\left(\frac{\left(\frac{2}{3}\right)^{2}}{2} \sin ^{-1} \frac{\left(x-\frac{1}{3}\right)}{\left(\frac{2}{3}\right)}+\frac{\left(x-\frac{1}{3}\right)}{2} \sqrt{\left(\frac{2}{3}\right)^{2}-\left(x-\frac{1}{3}\right)^{2}}\right)+c$
$y=\frac{2 \sqrt{3}}{9} \sin ^{-1} \frac{(3 x-1)}{2}+\frac{(3 x-1)}{6} \sqrt{1+2 x-3 x^{2}}+c$
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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,