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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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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
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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
How to Calculate the force given potential energy Read More »
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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Definite Integrals if $\int f(x) dx= g(x)$$\int_{a}^{b} f(x) dx =g(b) -g(a)$ Properties of Definite Integrals (I) $\int_{a}^{b} f(x) dx= \int_{a}^{b} f(t) dt $ (II) $\int_{a}^{b} f(x) dx=- \int_{b}^{a} f(x) dx $ (III) $\int_{a}^{b} f(x) dx= \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx $ where a<c <b (IV) $\int_{a}^{b} f(x) dx=\int_{a}^{b} f((a+b-x)) dx$ (V) $\int_{0}^{a} f(x)
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Integration of sin square x can be calculated using trigonometric identities .Here is the formula for it $$ \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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The integration of cot square x $\cot ^2 x$, can be found using trigonometry identities and fundamental integral formulas . The integral of $\cot^2 x$ with respect to (x) is: \[\int \cot^2 x \, dx =-\cot x -x + C\] Here, (C) represents the constant of integration, which is added because the process of integration
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