# 微積分の計算について　　　N0.4　定積分の内容－3　＜定積分の部分積分＞

###  学籍番号[＿＿＿＿＿＿＿＿＿]クラス[＿＿＿＿＿]　クラス番号[＿＿＿＿＿] 名前[＿＿＿＿＿＿＿＿＿＿＿＿＿＿＿]

##### 部分積分

###### 部分積分の公式　　$ [a,b]$で $f(x)$および $g'(x)$が連続で、$f(x)$の原始関数を$F(x)$とする

　$$   \int_{a}^{b} f(x)g(x)dx =  \left[ F(x)g(x) \right]_{a}^{b}- \int_{a}^{b} F(x)g'(x)dx $$

###### ワリスの公式（Wallis）
$$  \int_{0}^{\frac{\pi}{2}}\sin^n x dx = \int_{0}^{\frac{\pi}{2}}\cos^n x dx  $$
    n:奇数の時
    $$＝(\frac{n-1}{n})(\frac{n-3}{n-2})・・・(\frac{2}{3})(1) $$
    n:偶数の時
    $$＝(\frac{n-1}{n})(\frac{n-3}{n-2})・・・(\frac{1}{2})(\frac{\pi}{2}) $$


```python
from sympy import *
x, n , y, a = symbols('x n y a')
init_printing()
m ='4//3'
i =0
```


```python
expr = x*cos(x)
itg = Integral(expr,(x,0,pi/2))
i=i+1
print( 'No.',m,'---',i)
itg
```


```python
simplify(itg.doit())
```


```python
expr = x*exp(x)
itg = Integral(expr,(x,0,1))
i=i+1
print( 'No.',m,'---',i)
itg
```


```python
simplify(itg.doit())
```


```python
expr = x**2*exp(x)
itg = Integral(expr,(x,0,1))
i=i+1
print( 'No.',m,'---',i)
itg
```


```python
simplify(itg.doit())
```


```python
expr = exp(x)*cos(x)
itg = Integral(expr,(x,0,pi/2))
i=i+1
print( 'No.',m,'---',i)
itg
```


```python
simplify(itg.doit())
```


```python
expr =(log(x))**2
itg = Integral(expr,(x,1,exp(1)))
i=i+1
print( 'No.',m,'---',i)
itg
```


```python
simplify(itg.doit())
```


```python
expr = sin(x)**3
itg = Integral(expr,(x,0,pi/2))
i=i+1
print( 'No.',m,'---',i)
itg
```


```python
simplify(itg.doit())
```


```python
expr = cos(x)**4
itg = Integral(expr,(x,0,pi/2))
i=i+1
print( 'No.',m,'---',i)
itg
```


```python
simplify(itg.doit())
```


```python
expr = (x-1)**2*exp(-x)
itg = Integral(expr,(x,0,1))
i=i+1
print( 'No.',m,'---',i)
itg
```


```python
simplify(itg.doit())
```

361872ef59906e7910aa82a055cebd05d03119af
ipynb
Jupyter Notebook
08_20181121-sekibun-6-3-Ex&ans.ipynb
kt-pro-git-1/Calculus_Differential_Equation-public
d5deaf117e6841c4f6ceb53bc80b020220fd4814
MIT
2019-07-10T11:33:18.000Z
2019-07-10T11:33:18.000Z
08_20181121-sekibun-6-3-Ex&ans.ipynb
kt-pro-git-1/Calculus_Differential_Equation-public
d5deaf117e6841c4f6ceb53bc80b020220fd4814
MIT
08_20181121-sekibun-6-3-Ex&ans.ipynb
kt-pro-git-1/Calculus_Differential_Equation-public
d5deaf117e6841c4f6ceb53bc80b020220fd4814
MIT
Qwen/Qwen-72B
1. YES
2. YES