$\frac{\sin ^{-1} \sqrt{x}-\cos ^{-1} \sqrt{x}}{\sin ^{-1} \sqrt{x}+\cos ^{-1} \sqrt{x}}, x \in[0,1]$
Let $I=\int \frac{\sin ^{-1} \sqrt{x}-\cos ^{-1} \sqrt{x}}{\sin ^{-1} x+\cos ^{-1} x} d x$
It is known that, $\sin ^{-1} \sqrt{x}+\cos ^{-1} \sqrt{x}=\frac{\pi}{2}$
$\Rightarrow I=\int \frac{\left(\frac{\pi}{2}-\cos ^{-1} \sqrt{x}\right)-\cos ^{-1} \sqrt{x}}{\frac{\pi}{2}} d x$
$=\frac{2}{\pi} \int\left(\frac{\pi}{2}-2 \cos ^{-1} \sqrt{x}\right) d x$
$=\frac{2}{\pi} \cdot \frac{\pi}{2} \int 1 . d x-\frac{4}{\pi} \int \cos ^{-1} \sqrt{x} d x$
$=x-\frac{4}{\pi} \int \cos ^{-1} \sqrt{x} d x$ ...(1)
Let $I_{1}=\int \cos ^{-1} \sqrt{x} d x$
Also, let $\sqrt{x}=t \Rightarrow d x=2 t d t$
$\Rightarrow I_{1}=2 \int \cos ^{-1} t \cdot t d t$
$=2\left[\cos ^{-1} t \cdot \frac{t^{2}}{2}-\int \frac{-1}{\sqrt{1-t^{2}}} \cdot \frac{t^{2}}{2} d t\right]$
$=t^{2} \cos ^{-1} t+\int \frac{t^{2}}{\sqrt{1-t^{2}}} d t$
$=t^{2} \cos ^{-1} t-\int \frac{1-t^{2}-1}{\sqrt{1-t^{2}}} d t$
$=t^{2} \cos ^{-1} t-\int \sqrt{1-t^{2}} d t+\int \frac{1}{\sqrt{1-t^{2}}} d t$
$=t^{2} \cos ^{-1} t-\frac{t}{2} \sqrt{1-t^{2}}-\frac{1}{2} \sin ^{-1} t+\sin ^{-1} t$
$=t^{2} \cos ^{-1} t-\frac{t}{2} \sqrt{1-t^{2}}+\frac{1}{2} \sin ^{-1} t$
From equation (1), we obtain
$I=x-\frac{4}{\pi}\left[t^{2} \cos ^{-1} t-\frac{t}{2} \sqrt{1-t^{2}}+\frac{1}{2} \sin ^{-1} t\right]$
$=x-\frac{4}{\pi}\left[x \cos ^{-1} \sqrt{x}-\frac{\sqrt{x}}{2} \sqrt{1-x}+\frac{1}{2} \sin ^{-1} \sqrt{x}\right]$
$=x-\frac{4}{\pi}\left[x\left(\frac{\pi}{2}-\sin ^{-1}(\sqrt{x})\right)-\frac{\sqrt{x-x^{2}}}{2}+\frac{1}{2} \sin ^{-1}(\sqrt{x})\right]$
$=x-2 x+\frac{4 x}{\pi} \sin ^{-1} \sqrt{x}+\frac{2}{\pi} \sqrt{x-x^{2}}-\frac{2}{\pi} \sin ^{-1} \sqrt{x}$
$=-x+\frac{2}{\pi}\left[(2 x-1) \sin ^{-1} \sqrt{x}\right]+\frac{2}{\pi} \sqrt{x-x^{2}}+\mathrm{C}$
$=\frac{2(2 x-1)}{\pi} \sin ^{-1} \sqrt{x}+\frac{2}{\pi} \sqrt{x-x^{2}}-x+\mathrm{C}$
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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,