Khác biệt giữa các bản “Danh sách tích phân với hàm lượng giác”

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n (Đã lùi lại sửa đổi 12833857 của 222.255.199.80 (Thảo luận))
 
: <math>\int\sin ax\;dx = -\frac{1}{a}\cos ax+C\,\!</math>
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: <math>\int\sin^2 {ax}\;dx = \frac{x}{2} - \frac{1}{4a} \sin 2ax +C= \frac{x}{2} - \frac{1}{2a} \sin ax\cos ax +C\!</math>
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: <math>\int x\sin^2 {ax}\;dx = \frac{x^2}{4} - \frac{x}{4a} \sin 2ax - \frac{1}{8a^2} \cos 2ax +C\!</math>
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: <math>\int x^2\sin^2 {ax}\;dx = \frac{x^3}{6} - \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax - \frac{x}{4a^2} \cos 2ax +C\!</math>
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: <math>\int\sin b_1x\sin b_2x\;dx = \frac{\sin((b_1-b_2)x)}{2(b_1-b_2)}-\frac{\sin((b_1+b_2)x)}{2(b_1+b_2)}+C \qquad\mbox{(for }|b_1|\neq|b_2|\mbox{)}\,\!</math>
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: <math>\int\sin^n {ax}\;dx = -\frac{\sin^{n-1} ax\cos ax}{na} + \frac{n-1}{n}\int\sin^{n-2} ax\;dx \qquad\mbox{(for }n>2\mbox{)}\,\!</math>
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: <math>\int\frac{dx}{\sin ax} = \frac{1}{a}\ln \left|\tan\frac{ax}{2}\right|+C</math>
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: <math>\int\frac{dx}{\sin^n ax} = \frac{\cos ax}{a(1-n) \sin^{n-1} ax}+\frac{n-2}{n-1}\int\frac{dx}{\sin^{n-2}ax} \qquad\mbox{(for }n>1\mbox{)}\,\!</math>
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: <math>\int x\sin ax\;dx = \frac{\sin ax}{a^2}-\frac{x\cos ax}{a}+C\,\!</math>
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: <math>\int x^n\sin ax\;dx = -\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\;dx = \sum_{k=0}^{2k\leq n} (-1)^{k+1} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \cos ax +\sum_{k=0}^{2k+1\leq n}(-1)^k \frac{x^{n-1-2k}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \sin ax \qquad\mbox{(for }n>0\mbox{)}\,\!</math>
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: <math>\int_{\frac{-a}{2}}^{\frac{a}{2}} x^2\sin^2 {\frac{n\pi x}{a}}\;dx = \frac{a^3(n^2\pi^2-6)}{24n^2\pi^2} \qquad\mbox{(for }n=2,4,6...\mbox{)}\,\!</math>
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: <math>\int\frac{\sin ax}{x} dx = \sum_{n=0}^\infty (-1)^n\frac{(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!} +C\,\!</math>
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: <math>\int\frac{\sin ax}{x^n} dx = -\frac{\sin ax}{(n-1)x^{n-1}} + \frac{a}{n-1}\int\frac{\cos ax}{x^{n-1}} dx\,\!</math>
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: <math>\int\frac{dx}{1\pm\sin ax} = \frac{1}{a}\tan\left(\frac{ax}{2}\mp\frac{\pi}{4}\right)+C</math>
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: <math>\int\frac{x\;dx}{1+\sin ax} = \frac{x}{a}\tan\left(\frac{ax}{2} - \frac{\pi}{4}\right)+\frac{2}{a^2}\ln\left|\cos\left(\frac{ax}{2}-\frac{\pi}{4}\right)\right|+C</math>
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: <math>\int\frac{x\;dx}{1-\sin ax} = \frac{x}{a}\cot\left(\frac{\pi}{4} - \frac{ax}{2}\right)+\frac{2}{a^2}\ln\left|\sin\left(\frac{\pi}{4}-\frac{ax}{2}\right)\right|+C</math>
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