Evaluate the following integrals:
$\int \frac{\cos 2 x}{\sqrt{\sin ^{2} 2 x+8}} d x$
Let $=\sin 2 x$
$d t=2 \cos 2 x d x$
$\cos 2 x d x=d t / 2$
$\int \frac{\cos 2 \mathrm{x}}{\sqrt{\sin ^{2} 2 \mathrm{x}+8}} \mathrm{dx}=\frac{1}{2} \int \mathrm{dt} / \sqrt{\left(\mathrm{t}^{2}+(2 \sqrt{2})^{2}\right.}$
Since we have, $\int \frac{1}{\sqrt{\left(x^{2}+a^{2}\right)}} d x=\log \left[x+\sqrt{\left.\left(x^{2}+a^{2}\right)\right]+c}\right.$
$=\frac{1}{2} \int \mathrm{dt} / \sqrt{\left(\mathrm{t}^{2}+(2 \sqrt{2})^{2}\right.}=\frac{1}{2} \log \left[\mathrm{t}+\sqrt{\mathrm{t}^{2}+8}\right]+\mathrm{c}$
$=\frac{1}{2} \log \left[\mathrm{t}+\sqrt{\mathrm{t}^{2}+8}\right]+\mathrm{c}=\frac{1}{2} \log \left[\sin 2 \mathrm{x}+\sqrt{\sin ^{2} 2 \mathrm{x}+8}\right]+\mathrm{c}$
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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,