PART 3：常用的函數積分(06：54)

1. 最常見的積分函數首推多項式，但多項式並非只有下列這種
\((a)\int {{x^{10}}dx = \begin{array}{*{20}{c}}{}\\{\_\_\_\_\_\_\_\_\_\_\_}\end{array}} \quad ANS:\;\underline {\quad \frac{1}{{11}}{x^{11}} + C} \)
下列這些題型都是多項式的題型，各位同學要熟悉指數率
\((b)\int {\frac{1}{{{x^3}}}dx = \begin{array}{*{20}{c}}{}\\{\_\_\_\_\_\_\_\_\_\_\_}\end{array}} \quad ANS:\;\quad \underline { - \frac{1}{2}{x^{ - 2}} + C} \)
\((c)\int {\sqrt x dx = \begin{array}{*{20}{c}}{}\\{\_\_\_\_\_\_\_\_\_\_\_}\end{array}} \quad ANS:\;\underline {\quad \frac{2}{3}\sqrt {{x^3}}  + C} \)
\((d)\int {\frac{1}{x}dx = \begin{array}{*{20}{c}}{}\\{\_\_\_\_\_\_\_\_\_\_\_}\end{array}} \quad ANS:\;\quad \underline {\ln x + C\;} \quad x > 0\)
說明
以上例題都屬於積分法則4的狀況，依照下列說明，同學應該套用積分法則4輕易寫出正確答案
\((b)\;\frac{1}{{{x^3}}} = {x^{ - 3}}\)，

\((c)\ \ \sqrt{x}={{x}^{{}^{1}/{}_{2}}}\)，

\((d)\;\;n =  - 1\)是特殊狀況，答案是自然對數 \(\ln x\)
2. 自然指數與一般指數函數
\((e)\int {{e^x}dx = \begin{array}{*{20}{c}}{}\\{\_\_\_\_\_\_\_\_\_\_\_}\end{array}} \quad ANS:\;\quad {e^x} + C\;\)
\((f)\int {{a^x}dx = \begin{array}{*{20}{c}}{}\\{\_\_\_\_\_\_\_\_\_\_\_}\end{array}} \quad ANS:\;\quad \frac{{{a^x}}}{{\ln a}} + C\;\)
3. 三角函數
\((g)\int {\cos dx = \begin{array}{*{20}{c}}{}\\{\_\_\_\_\_\_\_\_\_\_\_}\end{array}} \quad ANS:\;\quad \underline {\sin x + C} \)
\((h)\int {\sin dx = \begin{array}{*{20}{c}}{}\\{\_\_\_\_\_\_\_\_\_\_\_}\end{array}} \quad ANS:\;\quad \underline { - \cos x + C} \)
\((i)\int {\sec x\tan xdx = \begin{array}{*{20}{c}}{}\\{\_\_\_\_\_\_\_\_\_\_\_}\end{array}} \quad ANS:\;\quad \underline {\sec x + C} \)
\((j)\int {{{\sec }^2}xdx = \begin{array}{*{20}{c}}{}\\{\_\_\_\_\_\_\_\_\_\_\_}\end{array}} \quad ANS:\;\quad \underline {\tan x + C} \)
4. 反三角相關.
\((k)\int {\frac{1}{{1 + {x^2}}}xdx = \begin{array}{*{20}{c}}{}\\{\_\_\_\_\_\_\_\_\_\_\_}\end{array}} \quad ANS:\;\quad \underline {Ta{n^{ - 1}}x + C} \)
\((l)\int {\frac{1}{{\sqrt {1 - {x^2}} }}xdx = \begin{array}{*{20}{c}}{}\\{\_\_\_\_\_\_\_\_\_\_\_}\end{array}} \quad ANS:\;\quad \underline {Si{n^{ - 1}}x + C} \)

要學好積分，看到以上這些不定積分，必須直接寫得出答案，才能培養解決更複雜的積分，
同學若知道答案從何而來，可以試著將答案微分看看並複習前面相關單元。