Evaluate the following integrals -
$\int x \sqrt{x^{2}+x} d x$
Let $I=\int x \sqrt{x^{2}+x} d x$
Let us assume $x=\lambda \frac{d}{d x}\left(x^{2}+x\right)+\mu$
$\Rightarrow x=\lambda\left[\frac{d}{d x}\left(x^{2}\right)+\frac{d}{d x}(x)\right]+\mu$
We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{n}}\right)=\mathrm{nx}^{\mathrm{n}-1}$
$\Rightarrow x=\lambda\left(2 x^{2-1}+1\right)+\mu$
$\Rightarrow x=\lambda(2 x+1)+\mu$
$\Rightarrow x=2 \lambda x+\lambda+\mu$
Comparing the coefficient of $x$ on both sides, we get
$2 \lambda=1 \Rightarrow \lambda=\frac{1}{2}$
Comparing the constant on both sides, we get
$\lambda+\mu=0$
$\Rightarrow \frac{1}{2}+\mu=0$
$\therefore \mu=-\frac{1}{2}$
Hence, we have $x=\frac{1}{2}(2 x+1)-\frac{1}{2}$
Substituting this value in I, we can write the integral as
$I=\int\left[\frac{1}{2}(2 x+1)-\frac{1}{2}\right] \sqrt{x^{2}+x} d x$
$\Rightarrow I=\int\left[\frac{1}{2}(2 x+1) \sqrt{x^{2}+x}-\frac{1}{2} \sqrt{x^{2}+x}\right] d x$
$\Rightarrow I=\int \frac{1}{2}(2 x+1) \sqrt{x^{2}+x} d x-\int \frac{1}{2} \sqrt{x^{2}+x} d x$
$\Rightarrow I=\frac{1}{2} \int(2 x+1) \sqrt{x^{2}+x} d x-\frac{1}{2} \int \sqrt{x^{2}+x} d x$
Let $I_{1}=\frac{1}{2} \int(2 x+1) \sqrt{x^{2}+x} d x$
Now, put $x^{2}+x=t$
$\Rightarrow(2 x+1) d x=d t$ (Differentiating both sides)
Substituting this value in $\mathrm{I}_{1}$, we can write
$\mathrm{I}_{1}=\frac{1}{2} \int \sqrt{\mathrm{t} \mathrm{dt}}$
$\Rightarrow \mathrm{I}_{1}=\frac{1}{2} \int \mathrm{t}^{\frac{1}{2}} \mathrm{dt}$
Recall $\int x^{n} d x=\frac{x^{n+1}}{n+1}+c$
$\Rightarrow \mathrm{I}_{1}=\frac{1}{2}\left(\frac{\mathrm{t}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right)+\mathrm{c}$
$\Rightarrow \mathrm{I}_{1}=\frac{1}{2}\left(\frac{\mathrm{t}^{\frac{3}{2}}}{\frac{3}{2}}\right)+\mathrm{c}$
$\Rightarrow I_{1}=\frac{1}{2} \times \frac{2}{3} t^{\frac{3}{2}}+c$
$\Rightarrow \mathrm{I}_{1}=\frac{1}{3} \mathrm{t} \frac{3}{2}+\mathrm{c}$
$\therefore \mathrm{I}_{1}=\frac{1}{3}\left(\mathrm{x}^{2}+\mathrm{x}\right)^{\frac{3}{2}}+\mathrm{c}$
Let $I_{2}=-\frac{1}{2} \int \sqrt{x^{2}+x} d x$
We can write $x^{2}+x=x^{2}+2(x)\left(\frac{1}{2}\right)+\left(\frac{1}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}$
$\Rightarrow x^{2}+x=\left(x+\frac{1}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}$
Hence, we can write $I_{2}$ as
$I_{2}=-\frac{1}{2} \int \sqrt{\left(x+\frac{1}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}} d x$
Recall $\int \sqrt{\mathrm{x}^{2}-\mathrm{a}^{2}} \mathrm{dx}=\frac{\mathrm{x}}{2} \sqrt{\mathrm{x}^{2}-\mathrm{a}^{2}}-\frac{\mathrm{a}^{2}}{2} \ln \left|\mathrm{x}+\sqrt{\mathrm{x}^{2}-\mathrm{a}^{2}}\right|+\mathrm{c}$
$\Rightarrow \mathrm{I}_{2}=-\frac{1}{2}\left[\frac{\left(\mathrm{x}+\frac{1}{2}\right)}{2} \sqrt{\left(\mathrm{x}+\frac{1}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}\right.$
$\left.-\frac{\left(\frac{1}{2}\right)^{2}}{2} \ln \left|\left(x+\frac{1}{2}\right)+\sqrt{\left(x+\frac{1}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}\right|\right]+c$
$\Rightarrow \mathrm{I}_{2}=-\frac{1}{2}\left[\frac{2 \mathrm{x}+1}{4} \sqrt{\mathrm{x}^{2}+\mathrm{x}}-\frac{1}{8} \ln \left|\mathrm{x}+\frac{1}{2}+\sqrt{\mathrm{x}^{2}+\mathrm{x}}\right|\right]+\mathrm{c}$
$\therefore \mathrm{I}_{2}=-\frac{1}{8}(2 \mathrm{x}+1) \sqrt{\mathrm{x}^{2}+\mathrm{x}}+\frac{1}{16} \ln \left|\mathrm{x}+\frac{1}{2}+\sqrt{\mathrm{x}^{2}+\mathrm{x}}\right|+\mathrm{c}$
Substituting $I_{1}$ and $I_{2}$ in $I$, we get
$I=\frac{1}{3}\left(x^{2}+x\right)^{\frac{3}{2}}-\frac{1}{8}(2 x+1) \sqrt{x^{2}+x}+\frac{1}{16} \ln \left|x+\frac{1}{2}+\sqrt{x^{2}+x}\right|+c$
Thus, $\int \mathrm{x} \sqrt{\mathrm{x}^{2}+\mathrm{x}} \mathrm{dx}=\frac{1}{3}\left(\mathrm{x}^{2}+\mathrm{x}\right)^{\frac{3}{2}}-\frac{1}{8}(2 \mathrm{x}+1) \sqrt{\mathrm{x}^{2}+\mathrm{x}}+\frac{1}{16} \ln \left|\mathrm{x}+\frac{1}{2}+\sqrt{\mathrm{x}^{2}+\mathrm{x}}\right|+\mathrm{c}$
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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,