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 $I=\int_{0}^{\pi/2} \log(\tan x) \, dx$ […]
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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 $I=\int_{0}^{\pi/2} \log(\cos x) \, dx$
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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: $$\int
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The integration of a periodic function involves some unique properties and approaches due to the function’s repeating nature. A function $f(x)$ is said to be periodic with period $T$ if for all $x$ in the domain of $f$, $f(x + T) = f(x)$. This periodicity can significantly simplify the integration process over intervals that are
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This can be done in these simple steps
1) First draw a free body diagram of the body showing all the forces on it.
2) Carefully choose a X-Y coordinate system so that direction of the motion or direction of force lies in X or Y direction. This will simplfy the Problem
3) Applying Newton’s law find out any unknown forces
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For one dimensional motion, the force can be found from Potential energy using following formula
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Two most common errors you should avoid when you see an average speed question.
Mistake #1
The most common mistakes in average speed problems occurs when the question asks you to calculate the average speed of a object moving at two different speeds for different parts of the journey. Students are generally tempted to calculate the arithmetic mean (average) of the two speeds, and select a corresponding answer. This averaging approach is wrong.
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Integration of sin square x can be calculated using trigonometric identities .Here is the formula for it $$ \int \sin^2(x) \, dx = \frac{1}{2} x – \frac{1}{4} \sin(2x) + C $$ Integration of sin 2x can be calculated using integration by substitutions .Here is the formula for it $$ \int \sin(2x) \, dx= \frac {-cos
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