微積分/不定積分/解答

1. 求解 $\int {\frac {3x}{2}}dx$

我們需要找到一個函式， $F$，使得

$F'(x)={\frac {3x}{2}}$

我們知道

${\frac {d}{dx}}x^{2}=2x$

所以我們需要找到一個常數， $a$，使得

${\frac {d}{dx}}ax^{2}=2ax={\frac {3x}{2}}$

解出 $a$，我們得到

$2ax={\frac {3x}{2}}\implies a={\frac {3}{4}}$

所以

$\int {\frac {3x}{2}}={\frac {3}{4}}x^{2}+C$

檢查你的答案，對找到的函式求導，並檢查它是否與被積函式匹配

${\frac {d}{dx}}\left({\frac {3}{4}}x^{2}+C\right)={\frac {3}{4}}(2x)={\frac {3x}{2}}$

2. 求函式 $f(x)=2x^{4}$ 的一般反導數。

我們知道

${\frac {d}{dx}}x^{5}=5x^{4}$
我們需要找到一個常數， $a$，使得
${\frac {d}{dx}}ax^{5}=5ax^{4}=2x^{4}$
解出 $a$，我們得到
$5ax^{4}=2x^{4}\implies a={\frac {2}{5}}$
因此，一般不定積分將為
$\mathbf {{\frac {2}{5}}x^{5}+C}$
透過對所求不定積分求導，並檢查是否回到初始函式，來驗證你的答案

${\frac {d}{dx}}\int 2x^{4}dx={\frac {d}{dx}}({\frac {2}{5}}x^{5}+C)={\frac {2}{5}}(5x^{4})=2x^{4}$

3. 求解 $\int (7x^{2}+3\cos(x)-e^{x})dx$

${\begin{aligned}\int (7x^{2}+3\cos(x)-e^{x})dx&=7\int x^{2}dx+3\int \cos(x)dx-\int e^{x}dx\\&=7({\frac {x^{3}}{3}})+3\sin(x)-e^{x}+C\\&=\mathbf {{\frac {7}{3}}x^{3}+3\sin(x)-e^{x}+C} \end{aligned}}$

4. 求解 $\int ({\frac {2}{5x}}+\sin(x))dx$

${\begin{aligned}\int ({\frac {2}{5x}}+\sin(x))dx&={\frac {2}{5}}\int {\frac {dx}{x}}+\int \sin(x)dx\\&=\mathbf {{\frac {2}{5}}\ln |x|-\cos(x)+C} \end{aligned}}$

5. 透過進行替換 $u=2x^{2}$，求解 $\int x\sin(2x^{2})dx$

因為 $u=2x^{2}$，$du=4xdx$ 並且 $dx={\frac {du}{4x}}$
${\begin{aligned}\int x\sin(2x^{2})dx&=\int x\sin(u){\frac {du}{4x}}\\&={\frac {1}{4}}\int \sin(u)du\\&=-{\frac {\cos(u)}{4}}+C\\&=-\mathbf {{\frac {\cos(2x^{2})}{4}}+C} \end{aligned}}$

6. 求解 $\int -3\cos(x)e^{\sin(x)}dx$

設 $u=\sin(x)$，$du=\cos(x)dx$，因此 $dx={\frac {du}{\cos(x)}}$
${\begin{aligned}\int -3\cos(x)e^{\sin(x)}dx&=-3\int \cos(x)e^{u}{\frac {du}{\cos(x)}}\\&=-3\int e^{u}du\\&=-3e^{u}+C\\&=\mathbf {-3e^{\sin(x)}+C} \end{aligned}}$

7. 使用分部積分法求解 $\int {\frac {2x-5}{x^{3}}}dx$，其中 $u=2x-5$ 以及 $dv={\frac {dx}{x^{3}}}$

$du=2dx$; $v=\int {\frac {dx}{x^{3}}}=-{\frac {1}{2x^{2}}}$
${\begin{aligned}\int {\frac {2x-5}{x^{3}}}dx&=\int udv\\&=uv-\int vdu\\&=(2x-5)(-{\frac {1}{2x^{2}}})-\int (-{\frac {1}{2x^{2}}})2dx\\&={\frac {5-2x}{2x^{2}}}+\int {\frac {dx}{x^{2}}}\\&={\frac {5-2x}{2x^{2}}}-{\frac {1}{x}}\\&={\frac {5-2x}{2x^{2}}}-{\frac {2x}{2x^{2}}}\\&=\mathbf {\frac {5-4x}{2x^{2}}} \end{aligned}}$

8. 計算 $\int (2x-1)e^{-3x+1}dx$

令 $u=2x-1$; $dv=e^{-3x+1}dx$

則 $du=2dx$ 且 $v=\int e^{-3x+1}dx$
為了計算 $v$, 進行以下代換 $w=-3x+1$; $dw=-3dx$; $dx={\frac {-dw}{3}}$. 那麼
$v=\int e^{-3x+1}dx=\int e^{w}({\frac {-1}{3}})dw={\frac {-e^{w}}{3}}={\frac {-e^{-3x+1}}{3}}$. 所以

${\begin{aligned}\int (2x-1)e^{-3x+1}dx&=\int udv\\&=uv-\int vdu\\&=(2x-1){\frac {-e^{-3x+1}}{3}}-\int {\frac {-e^{-3x+1}}{3}}(2)dx\\&={\frac {(1-2x)e^{-3x+1}}{3}}+{\frac {2}{3}}\int e^{-3x+1}dx\\&={\frac {(1-2x)e^{-3x+1}}{3}}+{\frac {2}{3}}\int {\frac {-e^{w}}{3}}dw\\&={\frac {3(1-2x)e^{-3x+1}}{9}}-{\frac {2}{9}}e^{w}\\&={\frac {(3-6x)e^{-3x+1}}{9}}-{\frac {2}{9}}e^{-3x+1}\\&=\mathbf {\frac {(1-6x)e^{-3x+1}}{9}} \end{aligned}}$