Question:
Evaluate the following integrals:
$\int \frac{\sin 8 x}{\sqrt{9+\sin ^{4} 4 x}} d x$
Solution:
Let $t=\sin ^{2} 4 x$
$d t=2 \sin 4 x \cos 4 x \times 4 d x$
we know $\sin 2 x=2 \sin 2 x \cos 2 x$
therefore, $d t=4 \sin 8 x d x$
or, $\sin 8 x d x=d t / 4$
$\int \frac{\sin 8 x}{\sqrt{9+\sin ^{4} x}} d x=\frac{1}{4} \int \frac{d t}{\sqrt{3^{2}+t^{2}}}$
Since we have, $\left.\int \frac{1}{\sqrt{\left(x^{2}+a^{2}\right)}} d x=\log \left[x+\sqrt{\left(x^{2}\right.}+a^{2}\right)\right]+c$
$=\frac{1}{4} \int \frac{d t}{\sqrt{3^{2}+t^{2}}}=\frac{1}{4} \log \left[t+\sqrt{t^{2}+3^{2}}+c\right.$
$=\frac{1}{4} \log \left[\sin ^{2} 4 x+\sqrt{9+\sin ^{4} 4 x}+c\right.$
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- 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,
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