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81.
cotxdx=lnsinx+C\int \cot x dx=\ln\left|\sin x\right|+C
82.
cosxsin2xdx=1sinx+C\int \dfrac{\cos x}{\sin^2x}dx=-\dfrac{1}{\sin x}+C
83.
cos2xsinxdx=lntanx2+cosx+C\int \dfrac{\cos^2 x}{\sin x}dx=\ln\left|\tan\dfrac{x}{2}\right|+cosx+C
84.
dxcosxsinx=lntanx+C\int \dfrac{dx}{\cos x \sin x}=\ln\left|\tan x\right|+C
85.
dxsin2xcosx=1sinx+lntan(x2+π2)+C\int \dfrac{dx}{\sin^2x \cos x} = - \dfrac{1}{\sin x}+\ln\left|\tan\left(\frac{x}{2}+\frac{\pi}{2}\right)\right|+C
86.
dxsinxcos2x=1cosc+lntanx2+C\int \dfrac{dx}{\sin x \cos^2x}=\dfrac{1}{\cos c}+\ln\left|\tan\frac{x}{2}\right|+C
87.
dxsin2xcos2x=tanxcotx+C\int \dfrac{dx}{\sin^2 x \cos^2x}=\tan x - \cot x + C
88.
sinmxsinnxdx=sin(m+n)x2(m+n)+sin(mn)x2(mn)+C,m2n2\int \sin mx \sin nx dx = -\dfrac{\sin(m+n)x}{2(m+n)}+\dfrac{\sin(m-n)x}{2(m-n)} + C, m^2 \ne n^2
89.
cosmxcosnxdx=sin(m+n)x2(m+n)+sin(mn)x2(mn)+C,m2n2\int \cos mx \cos nx dx = \dfrac{\sin(m+n)x}{2(m+n)}+\dfrac{\sin(m-n)x}{2(m-n)} + C, m^2 \ne n^2
90.
sinxcosnx=cosn+1xn+1+C\int \sin x \cos^nx= - \dfrac{\cos^{n+1}x}{n+1}+C
91.
sinnxcosx=sinn+1xn+1+C\int \sin^n x \cos x= \dfrac{\sin^{n+1}x}{n+1}+C
92.
Integral of arcsin
arcsinxdx=xarcsinx+1x2+C\int \arcsin x dx = x\arcsin x + \sqrt{1-x^2}+C
93.
sinmxcosnxdx=cos(m+n)x2(m+n)cos(mn)x2(mn)+C,m2n2\int \sin mx \cos nx dx = -\dfrac{\cos(m+n)x}{2(m+n)}-\dfrac{\cos(m-n)x}{2(m-n)} + C, m^2 \ne n^2
94.
Integral of arccos
arccosxdx=xarccosx1x2+C\int \arccos x dx = x\arccos x-\sqrt{1-x^2}+C
95.
Integral of arctan
arctanxdx=xarctanx12ln(x2+1)+C\int \arctan x dx = x\arctan x - \dfrac{1}{2}\ln\left(x^2+1\right)+C
96.
sin2α=1cos2α2\sin^2\alpha = \dfrac{1-\cos2\alpha}{2}
97.
sin3α=3sinαsin3α4\sin^3\alpha = \dfrac{3\sin\alpha - \sin 3\alpha}{4}
98.
sin4α=cos4α4cos2α+38\sin^4\alpha = \dfrac{\cos4\alpha-4\cos2\alpha+3}{8}
99.
sin5α=10sinα5sin3α+sin5α16\sin^5\alpha = \dfrac{10\sin\alpha - 5\sin3\alpha + \sin 5\alpha}{16}
100.
sin6α=1015cos2α+6cos4αcos6α32\sin^6\alpha = \dfrac{10-15\cos2\alpha+6\cos4\alpha-\cos6\alpha}{32}