Evaluate $\int(2 x+3) \sqrt{4 x^{2}+5 x+6} d x$
Make perfect square of quadratic equation
$4 x^{2}+5 x+6=4\left[\left(x+\frac{5}{8}\right)^{2}+\frac{71}{64}\right]$
$y=2 \int(2 x+3) \sqrt{\left[\left(x+\frac{5}{8}\right)^{2}+\left(\frac{\sqrt{71}}{8}\right)^{2}\right]} d x$
Let, $x+\frac{5}{8}=t \Rightarrow x=t-\frac{5}{8}$
Differentiate both side with respect to $t$
$\frac{d x}{d t}=1 \Rightarrow d x=d t$
$y=2 \int\left(2 t+\frac{7}{4}\right) \sqrt{\left[t^{2}+\left(\frac{\sqrt{71}}{8}\right)^{2}\right]} d t$
$A=4 \int t \sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+t^{2}} d t$
Let, $t^{2}=z$
Differentiate both side with respect to $z$
$2 t \frac{d t}{d z}=1 \Rightarrow t d t=\frac{1}{2} d z$
$A=2 \int \sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+z} d z$
$A=\frac{1}{4} \int \sqrt{71+64 z} d z$
$A=\frac{1}{384}(71+64 z)^{\frac{3}{2}}+c_{1}$
Put $z=t^{2}$ and $t=x+\frac{5}{8}$
$A=\frac{1}{384}\left(71+64\left(x+\frac{5}{8}\right)^{2}\right)^{\frac{3}{2}}+c_{1}$
$A=\frac{1}{6}\left(4 x^{2}+5 x+6\right)^{\frac{3}{2}}+c_{1}$
$B=\int \frac{7}{2} \sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+t^{2}} d t$
$B=\frac{7}{2}\left(\frac{t}{2} \sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+t^{2}+\frac{\left(\frac{\sqrt{71}}{8}\right)^{2}}{2}} \ln \left(t+\sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+t^{2}}\right)\right)+c_{2}$
Put $t=x+\frac{5}{8}$
$B=\frac{7}{2}\left(\frac{\left(x+\frac{5}{8}\right)}{2} \sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+\left(x+\frac{5}{8}\right)^{2}}\right)+$
$\frac{7\left(\frac{\sqrt{71}}{8}\right)^{2}}{4} \ln \left(\left(x+\frac{5}{8}\right)+\sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+\left(x+\frac{5}{8}\right)^{2}}\right)+c_{2}$
$B=\frac{7}{2}\left(\frac{(8 x+5)}{32} \sqrt{4 x^{2}+5 x+6}\right)+$
$\frac{497}{256} \ln \left(\left(x+\frac{5}{8}\right)+\sqrt{\left(\frac{\sqrt{71}}{8}\right)^{2}+\left(x+\frac{5}{8}\right)^{2}}\right)+c_{2}$
The final answer is $y=A+B$'
$y=\frac{1}{6}\left(4 x^{2}+5 x+6\right)^{\frac{3}{2}}+\frac{7}{2}\left(\frac{(8 x+5)}{32} \sqrt{4 x^{2}+5 x+6}\right)+$
$\frac{497}{256} \ln \left(\left(x+\frac{5}{8}\right)+\sqrt{x^{2}+\frac{5}{4} x+\frac{3}{2}}\right)+c$
$y=\frac{1}{192}\left(128 x^{2}+328 x+297\right) \sqrt{4 x^{2}+5 x+6}+$
$\frac{497}{256} \ln \left(\left(x+\frac{5}{8}\right)+\sqrt{x^{2}+\frac{5}{4} x+\frac{3}{2}}\right)+c$
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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,