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적분_공식 [2012/03/17 10:10] moonrepeat 새로 만듦 |
적분_공식 [2021/03/10 21:42] (현재) |
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| ^ 9 |$$ \int \frac{u}{\sqrt{a+bu}} \ du = \frac{2}{3b^{2}} (bu-2a) \sqrt{a+bu} + C $$ | | ^ 9 |$$ \int \frac{u}{\sqrt{a+bu}} \ du = \frac{2}{3b^{2}} (bu-2a) \sqrt{a+bu} + C $$ | | ||
| ^ 10 |$$ \int \frac{u^{2}}{\sqrt{a+bu}} \ du = \frac{1}{15b^{2}} (8a^{2}+3b^{2}u^{2}-4abu) \sqrt{a+bu} + C $$ | | ^ 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 |$$ \begin{displaymath}\begin{split} \int \frac{1}{u \sqrt{a+bu}} \ du &= \frac{1}{\sqrt{a}} \ln \left| \frac{\sqrt{a+bu} - \sqrt{a}}{\sqrt{a+bu} + \sqrt{a}} \right| + C \ \ , \ \operatorname{if} \ a > 0 \\ &= \frac{2}{\sqrt{-a}} \tan^{-1} \sqrt{\frac{a+bu}{-a}} + C \ \ \ \ , \ \operatorname{if} \ a < 0 \end{split}\end{displaymath} $$ | | + | ^ 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 |$$ \int \frac{\sqrt{a+bu}}{u} \ du = 2 \sqrt{a+bu} + a \ \int \frac{1}{u \sqrt{a+bu}} \ du $$ | | ^ 12 |$$ \int \frac{\sqrt{a+bu}}{u} \ du = 2 \sqrt{a+bu} + a \ \int \frac{1}{u \sqrt{a+bu}} \ du $$ | | ||
| ^ 13 |$$ \int \frac{\sqrt{a+bu}}{u^{2}} \ du = - \frac{\sqrt{a+bu}}{u} + \frac{b}{2} \ \int \frac{1}{u \sqrt{a+bu}} \ du $$ | | ^ 13 |$$ \int \frac{\sqrt{a+bu}}{u^{2}} \ du = - \frac{\sqrt{a+bu}}{u} + \frac{b}{2} \ \int \frac{1}{u \sqrt{a+bu}} \ du $$ | | ||
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| ^ 15 |$$ \int \frac{u^{n}}{\sqrt{a+bu}} \ du = \frac{2u^{n} \sqrt{a+bu}}{b(2n+1)} - \frac{2na}{b(2n+1)} \ \int \frac{u^{n-1}}{\sqrt{a+bu}} \ du $$ | | ^ 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 |$$ \int \frac{1}{u^{n} \sqrt{a+bu}} \ du = - \frac{\sqrt{a+bu}}{a(n-1) u^{n-1}} - \frac{b(2n-3)}{2a(n-1)} \ \int \frac{1}{u^{n-1} \sqrt{a+bu}} \ du $$ | | ^ 16 |$$ \int \frac{1}{u^{n} \sqrt{a+bu}} \ du = - \frac{\sqrt{a+bu}}{a(n-1) u^{n-1}} - \frac{b(2n-3)}{2a(n-1)} \ \int \frac{1}{u^{n-1} \sqrt{a+bu}} \ du $$ | | ||
| - | === Trigonmetric Forms === | + | ===== Trigonmetric Forms ===== |