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적분 공식
Basic Forms
| 1 | $$ \int u \ dv = uv - \int v \ du $$ | 2 | $$ \int u^{n} \ du = \frac{1}{n+1}u^{n+1} + C \ , \ n \not = -1 $$ |
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| 3 | $$ \int \frac{1}{u} \ du = \ln \ |u| + C $$ | 4 | $$ \int e^{u} \ du = e^{n} + C $$ |
| 5 | $$ \int a^{u} \ du = \frac{1}{\ln \ a} a^{u} + C $$ | 6 | $$ \int \sin \ u \ du = -\cos \ u + C $$ |
| 7 | $$ \int \cos \ u \ du = \sin \ u + C $$ | 8 | $$ \int \sec^{2} u \ du = \tan \ u + C $$ |
| 9 | $$ \int \csc^{2} u \ du = -\cot \ u + C $$ | 10 | $$ \int \sec \ u \ \tan \ u \ du = \csc \ u + C $$ |
| 11 | $$ \int \csc \ u \ \cot \ u \ du = -\csc \ u + C $$ | 12 | $$ \int \tan \ u \ du = \ln \ |\sec \ u| + C $$ |
| 13 | $$ \int \cot \ u \ du = \ln \ |\sin \ u| + C $$ | 14 | $$ \int \sec \ u \ du = \ln \ |\sec \ u + \tan \ u| + C $$ |
| 15 | $$ \int \csc \ u \ du = \ln \ |\csc \ u - \cot \ u| + C $$ | 16 | $$ \int \frac{1}{\sqrt{a^{2} - u^{2}}} \ du = \sin^{-1} \frac{u}{a} + C $$ |
| 17 | $$ \int \frac{1}{a^{2} + u^{2}} \ du = \frac{1}{a} \tan^{-1} \frac{u}{a} + C $$ | 18 | $$ \int \frac{1}{u \sqrt{u^{2}-a^{2}}} \ du = \frac{1}{a} \sec^{-1} \frac{u}{a} + C $$ |
| 19 | $$ \int \frac{1}{a^{2} - u^{2}} \ du = \frac{1}{2a} \ln \ |\frac{u + a}{u - a}| + C $$ | 20 | $$ \int \frac{1}{u^{2} - a^{2}} \ du = \frac{1}{2a} \ln \ |\frac{u - a}{u + a}| + C $$ |
Forms Involving 1
$$ \sqrt{a^{2} + u^{2}} \ , \ a > 0 $$
| 1 | $$ \int \sqrt{a^{2} + u^{2}} \ du = \frac{u}{2} \sqrt{a^{2} + u^{2}} + \frac{a^{2}}{2} \ln(u + \sqrt{a^{2} + u^{2}}) + C $$ |
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| 2 | $$ \int u^{2} \sqrt{a^{2} + u^{2}} \ du = \frac{u}{8} (a^{2} + 2u^{2}) \sqrt{a^{2} + u^{2}} - \frac{a^{4}}{8} \ln(u + \sqrt{a^{2} + u^{2}}) + C $$ |
| 3 | $$ \int \frac{\sqrt{a^{2} + u^{2}}}{u} \ du = \sqrt{a^{2} + u^{2}} - a \ \ln \ |\frac{a + \sqrt{a^{2} + u^{2}}}{u}| + C $$ |
| 4 | $$ \int \frac{\sqrt{a^{2} + u^{2}}}{u^{2}} \ du = - \frac{\sqrt{a^{2} + u^{2}}}{u} + \ln(u + \sqrt{a^{2} + u^{2}}) + C $$ |
| 5 | $$ \int \frac{1}{\sqrt{a^{2} + u^{2}}} \ du = \ln(u + \sqrt{a^{2} + u^{2}}) + C $$ |
| 6 | $$ \int \frac{u^{2}}{\sqrt{a^{2} + u^{2}}} \ du = \frac{u}{2} \sqrt{a^{2} + u^{2}} - \frac{a^{2}}{2} \ln (u + \sqrt{a^{2} + u^{2}}) + C $$ |
| 8 | $$ \int \frac{1}{u \sqrt{a^{2} + u^{2}}} \ du = - \frac{1}{a} \ln |\frac{\sqrt{a^{2} + u^{2}} + a}{u}| + C $$ |
| 9 | $$ \int \frac{1}{u^{2} \sqrt{a^{2} + u^{2}}} \ du = - \frac{\sqrt{a^{2} + u^{2}}}{a^{2} u} + C $$ |
| 10 | $$ \int \frac{1}{(a^{2} + u^{2})^{3/2}} \ du = \frac{u}{a^{2} \sqrt{a^{2} + u^{2}}} + C $$ |
Forms Involving 2
$$ \sqrt{a^{2} - u^{2}} \ , \ a > 0 $$
| 1 | $$ \int \sqrt{a^{2} - u^{2}} \ du = \frac{u}{2} \sqrt{a^{2} - u^{2}} + \frac{a^{2}}{2} \sin^{-1} \frac{u}{a} + C $$ |
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| 2 | $$ \int u^{2} \sqrt{a^{2} - u^{2}} \ du = \frac{u}{8} (2u^{2} - a^{2}) \sqrt{a^{2} - u^{2}} + \frac{a^{4}}{8} \sin^{-1} \frac{u}{a} + C $$ |
| 3 | $$ \int \frac{\sqrt{a^{2} - u^{2}}}{u} \ du = \sqrt{a^{2} - u^{2}} - a \ \ln|\frac{a + \sqrt{a^{2} - u^{2}}}{u}| + C$$ |
| 4 | $$ \int \frac{\sqrt{a^{2} - u^{2}}}{u^{2}} \ du = - \frac{1}{u} \sqrt{a^{2} - u^{2}} - \sin^{-1} \frac{u}{a} + C $$ |
| 5 | $$ \int \frac{u^{2}}{\sqrt{a^{2} - u^{2}}} \ du = - \frac{u}{2} \sqrt{a^{2} - u^{2}} + \frac{a^{2}}{2} \sin^{-1} \frac{u}{a} + C $$ |
| 6 | $$ \int \frac{1}{u \sqrt{a^{2} - u^{2}}} \ du = - \frac{1}{a} \ln |\frac{a + \sqrt{a^{2} - u^{2}}}{u}| + C $$ |
| 7 | $$ \int \frac{1}{u^{2} \sqrt{a^{2} - u^{2}}} \ du = - \frac{1}{a^{2} u} \sqrt{a^{2} - u^{2}} + C $$ |
| 8 | $$ \int (a^{2} - u^{2})^{3/2} \ du = - \frac{u}{8} (2u^{2} - 5a^{2}) \sqrt{a^{2} - u^{2}} + \frac{3a^{4}}{8} \sin^{-1} \frac{u}{a} + C $$ |
| 9 | $$ \int \frac{1}{(a^{2} - u^{2})^{3/2}} = \frac{u}{a^{2} \sqrt{a^{2} - u^{2}}} + C $$ |
Forms Involving 3
$$ \sqrt{u^{2} - a^{2}} \ , \ a > 0 $$
| 1 | $$ \int \sqrt{u^{2} - a^{2}} \ du = \frac{u}{2} \sqrt{u^{2} - a^{2}} - \frac{a^{2}}{2} \ln \ |u + \sqrt{u^{2} - a^{2}}| + C $$ |
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| 2 | $$ \int u^{2} \sqrt{u^{2} - a^{2}} \ du = \frac{u}{8} (2u^{2} - a^{2}) \sqrt{u^{2} - a^{2}} - \frac{a^{4}}{8} \ln \ |u + \sqrt{u^{2} - a^{2}}| + C $$ |
| 3 | $$ \int \frac{\sqrt{u^{2} - a^{2}}}{u} \ du = \sqrt{u^{2} - a^{2}} - a \ \cos^{-1} \frac{a}{|u|} + C $$ |
| 4 | $$ \int \frac{\sqrt{u^{2} - a^{2}}}{u^{2}} \ du = - \frac{\sqrt{u^{2} - a^{2}}}{u} + \ln \ |u + \sqrt{u^{2} - a^{2}}| + C $$ |
| 5 | $$ \int \frac{1}{\sqrt{u^{2} - a^{2}}} \ du = \ln \ |u + \sqrt{u^{2} - a^{2}}| + C $$ |
| 6 | $$ \int \frac{u^{2}}{\sqrt{u^{2} - a^{2}}} \ du = \frac{u}{2} \sqrt{u^{2} - a^{2}} + \frac{a^{2}}{2} \ln \ |u + \sqrt{u^{2} - a^{2}}| + C $$ |
| 7 | $$ \int \frac{1}{u^{2} \sqrt{u^{2} - a^{2}}} \ du = \frac{\sqrt{u^{2} - a^{2}}}{a^{2} u} + C $$ |
| 8 | $$ \int \frac{1}{(u^{2} - a^{2})^{3/2}} \ du = - \frac{u}{a^{2} \sqrt{u^{2} - a^{2}}} + C $$ |
Forms Involving 4
$$ a + bu $$
| 1 | $$ \int \frac{u}{a+bu} \ du = \frac{1}{b^{2}} (a+bu-a \ln |a+bu|) + C $$ | |||||||||||
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| 2 | $$ \int \frac{u^{2}}{a+bu} \ du = \frac{1}{2b^{3}}[(a+bu)^{2} - 4a(a+bu) + 2a^{2} \ln |a+bu|] + C $$ | |||||||||||
| 3 | $$ \int \frac{1}{u(a+bu)} \ du = \frac{1}{a} \ln \left| \frac{u}{a+bu} \right| + C $$ | |||||||||||
| 4 | $$ \int \frac{1}{u^{2}(a+bu)} \ du = -\frac{1}{au} + \frac{b}{a^{2}} \ln \left| \frac{u}{a+bu} \right| + C $$ | |||||||||||
| 5 | $$ \int \frac{u}{(a+bu)^{2}} \ du = \frac{a}{b^{2}(a+bu)} + \frac{1}{b^{2}} \ln |a+bu| + C $$ | |||||||||||
| 6 | $$ \int \frac{1}{u(a+bu)^{2}} \ du = \frac{1}{a(a+bu)} - \frac{1}{a^{2}} \ln \left| \frac{a+bu}{u} \right| +C $$ | |||||||||||
| 7 | $$ \int \frac{u^{2}}{(a+bu)^{2}} \ du = \frac{1}{b^{3}} \left( a+bu-\frac{a^{2}}{a+bu}-2a \ln |a+bu| \right) + C $$ | |||||||||||
| 8 | $$ \int u \sqrt{a+bu} \ du = \frac{1}{15b^{2}}(3bu-2a)(a+bu)^{3/2} + C $$ | |||||||||||
| 9 | $$ \int \frac{u}{\sqrt{a+bu}} \ du = \frac{2}{3b^{2}} (bu-2a) \sqrt{a+bu} + C $$ | |||||||||||
| 10 | $$ \int \frac{u^{2}}{\sqrt{a+bu}} \ du = \frac{1}{15b^{2}} (8a^{2}+3b^{2}u^{2}-4abu) \sqrt{a+bu} + C $$ | |||||||||||
| 11 | $$ \begin{displaymath}\begin{split} \int \frac{1}{u \sqrt{a+bu}} \ du &= \frac{1}{\sqrt{a}} \ln \left| \frac{\sqrt{a+bu} - \sqrt{a}}{\sqrt{a+bu} + \sqrt{a}} \right| + C \ \ , \ if \ a > 0 \\ &= \frac{2}{\sqrt{-a}} \tan^{-1} \sqrt{\frac{a+bu}{-a}} + C \ \ \ \ , \ if \ a < 0 \end{split}\end{displaymath} $$ | ^ 12 |$$ \int \frac{\sqrt{a+bu}}{u} \ du = 2 \sqrt{a+bu} + a \ \int \frac{1}{u \sqrt{a+bu}} \ du $$ | ^ 13 |$$ \int \frac{\sqrt{a+bu}}{u | {2}} \ du = - \frac{\sqrt{a+bu}}{u} + \frac{b}{2} \ \int \frac{1}{u \sqrt{a+bu}} \ du $$ | ^ 14 |$$ \int u | {n} \sqrt{a+bu} \ du = \frac{2}{b(2n+3)} \left[ u | {n} (a+bu) | {3/2} -na \ \int u | {n-1} \sqrt{a+bu} \ du \right] $$ | ^ 15 |$$ \int \frac{u | {n}}{\sqrt{a+bu}} \ du = \frac{2u | {n} \sqrt{a+bu}}{b(2n+1)} - \frac{2na}{b(2n+1)} \ \int \frac{u | {n-1}}{\sqrt{a+bu}} \ du $$ | ^ 16 |$$ \int \frac{1}{u | {n} \sqrt{a+bu}} \ du = - \frac{\sqrt{a+bu}}{a(n-1) u | {n-1}} - \frac{b(2n-3)}{2a(n-1)} \ \int \frac{1}{u | {n-1} \sqrt{a+bu}} \ du $$ |