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## Kinematics Good conceptual problems

Question: If you are riding on a train that speeds past other train moving in the same direction on adjacent train.It appears the other train is moving backward. Why?

Solution:
Your reference frame is that of the train you are riding. If you are traveling with a relatively constant velocity (not over

## Mechanics of Turning a car on the curve

Since the car is turning on a curve, The car will be accelerating as direction of the car changes even the velocity is constant. This acceleration is centripetal acceleration And centripetal force will be providing it,
There are three cases with turning
1) Unbanked curve.
2) Banked curve with no friction
3) Banked curve with friction

## Integration of log tanx

For integration of $\log(\tan x)$, we generally consider the definite integral over the interval from 0 to $\pi/2$ To calculate the definite integral of $\log(\tan x)$ from (0) to $\pi/2$, we use a technique involving symmetry and the properties of logarithms. The integral is: $\int_{0}^{\pi/2} \log(\tan x) \, dx$ Let $I=\int_{0}^{\pi/2} \log(\tan x) \, dx$

## integration of log cosx

For integration of $\log(\cos x)$, we generally consider the definite integral over the interval from 0 to $\pi/2$ To calculate the definite integral of $\log(\cos x)$ from (0) to $\pi/2$, we use a technique involving symmetry and the properties of logarithms. The integral is: $\int_{0}^{\pi/2} \log(\cos x) \, dx$ Let $I=\int_{0}^{\pi/2} \log(\cos x) \, dx$

## integration of e^{ax} sin bx

The integral ($\int e^{ax} \sin(bx) \, dx$) is computed as: $\int e^{ax} \sin(bx) \, dx= \frac{a e^{ax} \sin(bx)}{a^2 + b^2} – \frac{b e^{ax} \cos(bx)}{a^2 + b^2}+ C$ Proof of Integration To solve the integral $\int e^{ax} \sin(bx) \, dx$, we can use the method of integration by parts, which is based on the formula: \int

## Integration of periodic functions

The integration of a periodic function involves some unique properties and approaches due to the function’s repeating nature. A function $f(x)$ is said to be periodic with period $T$ if for all $x$ in the domain of $f$, $f(x + T) = f(x)$. This periodicity can significantly simplify the integration process over intervals that are

## Integration of odd function

The integration of an odd function over a symmetric interval can be understood through some fundamental concepts in calculus and symmetry. An odd function is defined as a function $f(x)$ that satisfies the condition $f(-x) = -f(x)$ for all $x$ in the domain of the function. Graphically, odd functions exhibit symmetry about the origin, meaning

## How to find work done by Multiple forces acting on a object

This can be done in these simple steps

1) First draw a free body diagram of the body showing all the forces on it.

2) Carefully choose a X-Y coordinate system so that direction of the motion or direction of force lies in X or Y direction. This will simplfy the Problem

3) Applying Newton’s law find out any unknown forces