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: $$\int […]
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The integration of a periodic function involves some unique properties and approaches due to the function’s repeating nature. A function $f(x)$ is said to be periodic with period $T$ if for all $x$ in the domain of $f$, $f(x + T) = f(x)$. This periodicity can significantly simplify the integration process over intervals that are
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Integration of sin square x can be calculated using trigonometric identities .Here is the formula for it $$ \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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The integration of tan square x $\tan ^2 x$, can be found using trigonometry identities and fundamental integral formulas . The integral of $\tan ^2 x$ with respect to (x) is: \[\int \tan^2 x \, dx =\tan x -x + C\] Here, (C) represents the constant of integration, which is added because the process of
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The integration of sin cube x $\sin ^3 x$, can be found using integration substitution and trigonometry identities . The integral of $\sin ^3 x$ with respect to (x) is: \[\int \sin^3 x \, dx =\frac {\cos 3x}{12} – \frac {3\cos x}{4} + C\] or \[\int \sin^3 x \, dx =+\frac{\cos^3(x)}{3} – \cos(x) + C\]
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The differentiation of (\cos(x)) with respect to (x) is a fundamental concept in calculus. The derivative of (\cos(x)) is given by: \[\frac{d}{dx}(\cos(x)) = -\sin(x).\] This result is derived from the limit definition of the derivative and is a key part of the differentiation rules for trigonometric functions. The negative sine function, $-\sin(x)$, represents the rate
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Number 54 is a composite number and we will find how to find the factors of 54. We will also see techniques to find out the Prime factorization of 54 easily Factors of 54 A factor of a number is an exact divisor of that number. So factors of 54 are the numbers which are
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