```python
from IPython.display import Image
from IPython.core.display import HTML 
from sympy import *
Image(url= "https://i.imgur.com/4HeGIe7.png")
```









```python
t = symbols("t")
Integral(sqrt(4*t+9))
```




$\displaystyle \int \sqrt{4 t + 9}\, dt$




```python
simplify(Integral(sqrt(4*t+9)).doit())

```




$\displaystyle \frac{\left(4 t + 9\right)^{\frac{3}{2}}}{6}$




```python
Image(url= "https://i.imgur.com/tAO2Ij3.png")

```









```python
Image(url= "https://i.imgur.com/kXlWcEb.png")
```









```python
x = symbols("x")
Integral(x**5*sin(x**6))
```




$\displaystyle \int x^{5} \sin{\left(x^{6} \right)}\, dx$




```python
print(Integral(x**5*sin(x**6)).doit())
```

    -cos(x**6)/6



```python
Image(url= "https://i.imgur.com/Mr3KaY2.png")
```









```python
Image(url= "https://i.imgur.com/RpUQmc0.png")
```









```python
from sympy import pi
Integral(cos(pi/x)/x**2)
```




$\displaystyle \int \frac{\cos{\left(\frac{\pi}{x} \right)}}{x^{2}}\, dx$




```python
print(Integral(cos(pi/x)/x**2).doit())
```

    -sin(pi/x)/pi



```python
Image(url= "https://i.imgur.com/Nj4pFIO.png")
```









```python
Image(url= "https://i.imgur.com/u7LsgcI.png")
```









```python
Integral(sin(x)**4*cos(x))
```




$\displaystyle \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$




```python
print(Integral(sin(x)**4*cos(x)).doit())
```

    sin(x)**5/5



```python
Image(url= "https://i.imgur.com/Bytu7tM.png")
```









```python
Image(url= "https://i.imgur.com/ZTp8zyi.png")
```









```python
Integral(x*cos(1-x**2))
```




$\displaystyle \int x \cos{\left(x^{2} - 1 \right)}\, dx$




```python
print(Integral(x*cos(1-x**2)).doit())
```

    sin(x**2 - 1)/2



```python
Image(url= "https://i.imgur.com/D72UOm8.png")
```









```python
Image(url= "https://i.imgur.com/Tt4wiJA.png")
```









```python
(Integral(cos(4*t)**15*sin(4*t)))
```




$\displaystyle \int \sin{\left(4 t \right)} \cos^{15}{\left(4 t \right)}\, dt$




```python
print(Integral(cos(4*t)**15*sin(4*t)).doit())
```

    -cos(4*t)**16/64



```python
Image(url= "https://i.imgur.com/fp90pIZ.png")
```









```python
Image(url= "https://i.imgur.com/87ipBxf.png")
```









```python
Integral(sec(5*x)*tan(5*x))
```




$\displaystyle \int \tan{\left(5 x \right)} \sec{\left(5 x \right)}\, dx$




```python
print(Integral(sec(5*x)*tan(5*x)).doit())
```

    1/(5*cos(5*x))



```python
Image(url= "https://i.imgur.com/p3TV8lD.png")
```









```python
Image(url= "https://i.imgur.com/VOZPhlq.png")
```









```python
Integral((2*x-5)**5,(x,3,5))
```




$\displaystyle \int\limits_{3}^{5} \left(2 x - 5\right)^{5}\, dx$




```python
print(Integral((2*x-5)**5,(x,3,5)).doit())
```

    1302



```python
Image(url= "https://i.imgur.com/IEjf5cG.png")
```









```python
Image(url= "https://i.imgur.com/SSN6ZNW.png")
```









```python
Integral(4/(sqrt(9-4*x)),(x,-4,-10))
```




$\displaystyle \int\limits_{-4}^{-10} \frac{4}{\sqrt{9 - 4 x}}\, dx$




```python
print(Integral(4/(sqrt(9-4*x)),(x,-4,-10)).doit())
```

    -4



```python
Image(url= "https://i.imgur.com/GpwBaxw.png")
```









```python
Image(url= "https://i.imgur.com/McCH6Kx.png")
```









```python
Integral(8*x**9 + 8*x**4)
```




$\displaystyle \int \left(8 x^{9} + 8 x^{4}\right)\, dx$




```python
print(Integral(8*x**9 + 8*x**4).doit())
```

    4*x**10/5 + 8*x**5/5



```python
Image(url= "https://i.imgur.com/Hziq0U9.png")
```









```python
Image(url= "https://i.imgur.com/cMQDWXn.png")
```









```python
from sympy import Function
f = Function('f')
def f(x):
    return x
Eq(Integral(f(x),(x,0,4)),8)
```




$\displaystyle \int\limits_{0}^{4} x\, dx = 8$




```python
Integral(f(2*x),(x,0,2)).doit()
```




$\displaystyle 4$




```python
Image(url= "https://i.imgur.com/Ir5yAmw.png")
```









```python
Image(url= "https://i.imgur.com/hIFtNsa.png")
```









```python
Integral(sqrt(5-2*x),(x,-4,2))
```




$\displaystyle \int\limits_{-4}^{2} \sqrt{5 - 2 x}\, dx$




```python
print(Integral(sqrt(5-2*x),(x,-4,2)).doit())
```

    -1/3 + 13*sqrt(13)/3



```python
Image(url= "https://i.imgur.com/RXi96fW.png")
```









```python
Image(url= "https://i.imgur.com/Kg571fy.png")
```









```python
e5 = sin(x**(Rational(1,4)))/x**(Rational(3,4))
e5
```




$\displaystyle \frac{\sin{\left(\sqrt[4]{x} \right)}}{x^{\frac{3}{4}}}$




```python
e6 = Integral(e5,(x,1,16)).doit()
e6
```




$\displaystyle - 4 \cos{\left(2 \right)} + 4 \cos{\left(1 \right)}$




```python
print(e6)
```

    -4*cos(2) + 4*cos(1)



```python
Image(url= "https://i.imgur.com/IZerZus.png")
```









```python
Image(url= "https://i.imgur.com/0JLEO9z.png")
```









```python
e7 = 3*sin(x)/cos(x)**2
e7
```




$\displaystyle \frac{3 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$




```python
Integral(e7,(x,0,pi/3))
```




$\displaystyle \int\limits_{0}^{\frac{\pi}{3}} \frac{3 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}\, dx$




```python
Integral(e7,(x,0,pi/3)).doit()
```




$\displaystyle 3$




```python
Image(url= "https://i.imgur.com/ZN10AB4.png")
```









```python
Image(url= "https://i.imgur.com/mb5y8z8.png")
```









```python
Integral(cos(8*x)*sqrt(14-sin(8*x)))
```




$\displaystyle \int \sqrt{14 - \sin{\left(8 x \right)}} \cos{\left(8 x \right)}\, dx$




```python
print(Integral(cos(8*x)*sqrt(14-sin(8*x))).doit())
```

    sqrt(14 - sin(8*x))*sin(8*x)/12 - 7*sqrt(14 - sin(8*x))/6



```python
Image(url= "https://i.imgur.com/E2evsbD.png")
```









```python
Image(url= "https://i.imgur.com/iFHeHfX.png")
```









```python
Integral(1/x**5*sqrt(1+(1/x**4)),(x,1,4))
```




$\displaystyle \int\limits_{1}^{4} \frac{\sqrt{1 + \frac{1}{x^{4}}}}{x^{5}}\, dx$




```python
print(Integral(1/x**5*sqrt(1+(1/x**4)),(x,1,4)).doit())
```

    -257*sqrt(257)/24576 + sqrt(2)/3



```python
Image(url= "https://i.imgur.com/nceIFbp.png")
```









```python
Image(url= "https://i.imgur.com/NxrFbm7.png")
```









```python
Integral(x**2*(1+3*x**3)**5,(x,0,1))
```




$\displaystyle \int\limits_{0}^{1} x^{2} \left(3 x^{3} + 1\right)^{5}\, dx$




```python
print(Integral(x**2*(1+3*x**3)**5,(x,0,1)).doit())
```

    455/6



```python
Image(url= "https://i.imgur.com/RXX2ruO.png")
```









```python
Image(url= "https://i.imgur.com/8u9zXhU.png")
```









```python
Integral((4*x**3)/(x**4+1)**2,(x,2,4))
```




$\displaystyle \int\limits_{2}^{4} \frac{4 x^{3}}{\left(x^{4} + 1\right)^{2}}\, dx$




```python
print(Integral((4*x**3)/(x**4+1)**2,(x,2,4)).doit())

```

    240/4369



```python
Image(url= "https://i.imgur.com/pqFk6Hu.png")
```









```python
Image(url= "https://i.imgur.com/vFmjBoG.png")
```









```python
y = symbols('y')
Integral(y*sqrt(y+8))
```




$\displaystyle \int y \sqrt{y + 8}\, dy$




```python
print(Integral(y*sqrt(y+8)).doit())
```

    2*y**2*sqrt(y + 8)/5 + 16*y*sqrt(y + 8)/15 - 256*sqrt(y + 8)/15



```python
Image(url= "https://i.imgur.com/Ef9IGex.png")
```









```python
Image(url= "https://i.imgur.com/iniy2ix.png")
```









```python
f = 12/(4*x-7)**2
f
```




$\displaystyle \frac{12}{\left(4 x - 7\right)^{2}}$




```python
g = Integral(f)
c = symbols('c')
Eq(g.subs(x,2)+c,-6)
```




$\displaystyle c + \int\limits^{2} \frac{12}{\left(4 x - 7\right)^{2}}\, dx = -6$




```python
solve(Eq(g.subs(x,2)+c,-6),c)
```




    [-3]




```python
print(g.doit()-3)
```

    -3 - 12/(16*x - 28)



```python
Image(url= "https://i.imgur.com/K2tpqzf.png")
```








c272efbd920eb26a6875730a95c4a68d1c64559e
ipynb
Jupyter Notebook
Calculus_Homework/WWB20.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Calculus_Homework/WWB20.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Calculus_Homework/WWB20.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Qwen/Qwen-72B
1. YES
2. YES