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We can easily find the integration of $\frac {1}{1+ cos x}$ and $\frac {1}{1 -cos x}$ using trigonometric identities and fundamental integration techniques, The formula of these integral is given as $\int \frac {1}{1+ cos x} \; dx = \tan \frac {x}{2} + C$ or $\int \frac {1}{1+ cos x} \; dx = -\cot x […]
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The integration of tan cube x $\tan ^3 x$, can be found using integration substitution and trigonometry identities . The integral of $\tan ^3 x$ with respect to (x) is: \[\int \tan^3 x \, dx =\frac {1}{2} \tan^2 x – \ln |sec x|+ C\] Here, (C) represents the constant of integration, which is added because
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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: $$\int
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The integration of a periodic function involves some unique properties and approaches due to the function’s repeating nature. A function $f(x)$ is said to be periodic with period $T$ if for all $x$ in the domain of $f$, $f(x + T) = f(x)$. This periodicity can significantly simplify the integration process over intervals that are
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Integration of sin square x can be calculated using trigonometric identities .Here is the formula for it $$ \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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