Question:
Evaluate $\int \cos ^{5} x d x$.
Solution:
$y=\int\left(1-\sin ^{2} x\right)^{2} \cos x d x$
Let, $\sin x=t$
Differentiating both side with respect to $x$
$\frac{d t}{d x}=\cos x \Rightarrow d t=\cos x d x$
$y=\int\left(1-t^{2}\right)^{2} d t$
$y=\int 1+t^{4}-2 t^{2} d t$
Using formula $\int t^{n} d t=\frac{t^{n+1}}{n+1}$ and $\int c d t=c t$
$y=\left(t+\frac{t^{5}}{5}-2 \frac{t^{3}}{3}\right)+c$
Again, put $t=\sin x$
$y=\left(\sin x+\frac{\sin ^{5} x}{5}-2 \frac{\sin ^{3} x}{3}\right)+c$
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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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