# Initail Value Problem
> ## Given $\frac{dy}{dx} = 8x + csc^2(x)$ with $y(\pi/2) = -7$ solve for $y(x)$


```python
import sympy as sm
x, C = sm.symbols('x C')
sol = sm.integrate(8*x + sm.csc(x)**2, x)

# y = sol + C
# y(pi/2) = -7

C = -sol - 7
ans = sol + C.subs(x,sm.pi/2)
ans
# ans.subs(x,sm.pi/2)
```




$\displaystyle 4 x^{2} - \pi^{2} - 7 - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$



> ## $$ \int \frac{(1+\sqrt{x})^{1/3}}{\sqrt{x}}dx$$


```python
(1 + sm.sqrt(x))**sm.Rational(1,3) / sm.sqrt(x)
```




$\displaystyle \frac{\sqrt[3]{\sqrt{x} + 1}}{\sqrt{x}}$



> ## $$
\int x(1-x^2)^{1/4} dx
$$


```python
(x*(1-x**2))**sm.Rational(1/4)
```




$\displaystyle \sqrt[4]{x \left(1 - x^{2}\right)}$



# Definite Integrals
> ## $$ \int_0^{ln(4)}\frac{e^x}{e^{2x}+9}dt $$


```python
sm.exp(x) / (sm.exp(2*x) + 9)
```




$\displaystyle \frac{e^{x}}{e^{2 x} + 9}$



> ## $$\int _1^t x^{10}e^x dx $$


```python
t = sm.symbols('t')
sm.integrate(x**10 * sm.exp(x),(x,1,t))
```




$\displaystyle \left(t^{10} - 10 t^{9} + 90 t^{8} - 720 t^{7} + 5040 t^{6} - 30240 t^{5} + 151200 t^{4} - 604800 t^{3} + 1814400 t^{2} - 3628800 t + 3628800\right) e^{t} - 1334961 e$



# Improper Integrals
> ## $$ \int_0^{\infty} \frac{16tan^-1(x)}{1+x^2}dx$$


```python
sm.integrate(16*sm.atan(x) / (1+x**2),(x,0,sm.oo))
```




$\displaystyle 2 \pi^{2}$




```python

```

88879a26d06acff39aaefab30d90c747c2cc72d3
ipynb
Jupyter Notebook
python/Vectors/ivp.ipynb
karng87/nasm_game
a97fdb09459efffc561d2122058c348c93f1dc87
MIT
python/Vectors/ivp.ipynb
karng87/nasm_game
a97fdb09459efffc561d2122058c348c93f1dc87
MIT
python/Vectors/ivp.ipynb
karng87/nasm_game
a97fdb09459efffc561d2122058c348c93f1dc87
MIT
Qwen/Qwen-72B
1. YES
2. YES