```python
from sympy import *
from sympy.abc import x, y, e
init_printing(use_unicode=True)

```


```python
a1 = (x**7 + sin(x))
display(a1)
display(a1.diff())
```


```python
a2 = sqrt(x)
display(a2)
display(a2.diff())
```


```python
a3 = e**(3*x)*sin(x)
display(a3)
display(a3.diff(x).factor())
```


```python
a4 = Eq(sqrt(x) + sqrt(y), 1)
display(a4)
display(Derivative(a4, y, x))
```


```python
a5 = (2*x**4 + 3*x**2 + x + 8)
display(a5)
display(integrate(a5))
```


```python
a6 = (sqrt(x) + 1/sqrt(x))
display(a6)
display(integrate(a6))
```


```python
# Warning, this is a burn processor integral :

a7 = (x**3*(1-12*x**4))**Rational(1/8)
display(a7)
display(integrate(a7))
```


```python
a8 = (x/(sqrt(8-2*x**2)))
display(a8)
display(integrate(a8))
```


```python
a10 = sin(5*x)
display(a10)
display(integrate(a10))
```


```python
a11 = cos(x)*(sin(x)**3)**-1
display(a11)
display(integrate(a11))
```


```python
a12 = tan(x)**6 * sec(x)**2
display(a12)
res = integrate(a12)
display(res)
display(integrate(a12).simplify())
```


```python
(x**3-x).diff()
```


```python
k = 1
n = 1000
r = sum(pow(-1, i) for i in range(k,n))
print(r)
```

    -1



```python
x, k = symbols("x k")
s = Sum(Indexed('-1',k),(k,1,1001))
display(s)
```


```python
# primes, negative-positive iteration
x, k = symbols("x k")
s = Sum(Indexed('(-1)^n+1 * 2*n+1',k),(k,1,20))
display(s)

for p in range(1, 20):
    print( ((-1)**(p+1)) * ((2*p) + 1) )

```


```python

```

c6578a9ca637e63c6faf3895eb0ebe1d290ff215
ipynb
Jupyter Notebook
problem_set_1_part_a/lab.ipynb
beltrewilton/mitx_18.01.2x
5b3b260b9ef979b17d28465824e98a5138a9e8b6
MIT
problem_set_1_part_a/lab.ipynb
beltrewilton/mitx_18.01.2x
5b3b260b9ef979b17d28465824e98a5138a9e8b6
MIT
problem_set_1_part_a/lab.ipynb
beltrewilton/mitx_18.01.2x
5b3b260b9ef979b17d28465824e98a5138a9e8b6
MIT
Qwen/Qwen-72B
1. YES
2. YES