## Jgeurts 2 years ago Evaluate the following integrals. Show your work.

1. zepp

1/cos^2(x) is sec^2(x) and the integral of that is tan(x)

2. Jgeurts

$\int\limits_{0}^{2}x \sqrt{9-2x^2}dx$

3. Jgeurts

$\frac{ d }{ dx }\int\limits_{x^2}^{sinx}\sqrt{1+t^2}dt$

4. primeralph

no, just substitute, then everything falls out

5. Best_Mathematician

use calculator so easy

6. Jgeurts

I need to do it without @best_mathematician

7. Best_Mathematician

ok here...

8. modphysnoob

integral and derivative cancel each other out

9. Best_Mathematician

now u know wht to do right @Jgeurts

10. Jgeurts

yes thanks BM

11. Jgeurts

12. modphysnoob

$\frac{ d }{ dx }\int\limits_{x^2}^{sinx}\sqrt{1+t^2}dt$ after our supposed integration F(Sin(x)) - F ( x^2) take dervaitve d/dx of F( Sin(x)) = cos(x) f(sin(x) d/dx of F(x^2) = f (x^2) 2x go back and plug into your funciton

13. modphysnoob

f(t)= Sqrt[1+t^2] f(sin(x)= sqrt[1+ sin^2(x)] f(sin(x)*cos(x)= sqrt[1+ sin^2(x)]* cos(x)

14. modphysnoob

now the bottom part f(x^2)2x=sqrt[1+x^2]2x

15. modphysnoob

sqrt[1+ sin^2(x)]* cos(x)-sqrt[1+x^2]2x

16. modphysnoob

question?

17. Jgeurts

Im formulating my question, haha

18. modphysnoob

ok

19. Jgeurts

so the d/dx means derive the interval?

20. modphysnoob

yes, since we are taking a function integrate it and derive , it undoes the integration

21. Jgeurts

haha great! i get it, thank you, that one was super hard!

22. modphysnoob

I can show you an easy example if you like?

23. Jgeurts

@modphysnoob sure that would be great, im learning for my final :)

24. modphysnoob

let's to a problem like yours but with easier integrand |dw:1367723123193:dw|

25. Jgeurts

so we would derive sinx to cosx and x2 to 2x

26. Jgeurts

then plug in for t and subtract them from each other right?

27. Jgeurts

so cosx-2x?

28. modphysnoob

so in this case f(t)= t when we integrate we will get a function which we will call F(t) F(t) is evaluated at F(sin(x)) - F( x^2) then taken d/dx of d/dx( F(sin(x)) - F( x^2) ) using chain rule d/dx ( F(sin(x)) = cos(x) f( sin(x)

29. modphysnoob

which is cos(x) sin(x)

30. Jgeurts

i see

31. modphysnoob

let's do botttom part d/dx( F (x^2) = 2x f( x^2) 2x x^2= 2x^3

32. modphysnoob

so cos(x) sin(x)-2x^3

33. modphysnoob

this is easy integrand so we could actually integrate it and then differntiate it to confirm out result. Integrate t t^2/2 plug in limits (sin^2 x)/2 - x^4/2 take derivative d/dx cos(x) sin(x)- 2x^3

34. modphysnoob

got it?

35. Jgeurts

Yes thats great! thank you so much! @modphysnoob

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