[Help] Integral Problem

Ok I cant figure out how to solve this. I should be able to just plug in the sin(x) into the t's but it says that is the wrong answer. I asked my prof in class today to solve it, but he really didnt help. So maybe you guys can help me out. Thanks in advanced.

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StatusX
Homework Helper
What does the fundamental theorem of calculus say? Also, you'll need to use the chain rule.

Its says the derivative of g(x) is equal f(x), I know the Fundamental Theorem of Calculus, but I just dont know how to use it alongside the chain rule. Our book only has one example of it, and it doesnt explain it very well.

I think the answer is 5x^4*cos(x^5)*cos(sin(x)^5)+sin(x) but, I know its wrong cus the website wont take it.

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StatusX
Homework Helper
Yea, but that's meaningless unless you define f(x) and g(x). I'm sure you do know it, but all this problem consists of is carefully applying the definitions, so you should be explicit.

HallsofIvy
Homework Helper
waffle07 said:
Its says the derivative of g(x) is equal f(x)
No matter what f and g are? What a remarkable theorem!:rofl:

One form of the fundamental theorem of calculus says that if
$$g(x)= \int_{x_0}^x f(t)dt$$
then the derivative of g(x) if f(x). Here your only "problem" is that the upper limit is sin(x) instead of x. Can you make a substitution to correct that? As StatusX says, use the chain rule.

Alright that didnt help me at all. Um, lets see can you just tell me the answer?!?! I think I would be able to see what Im doing wrong if I knew the answer.

StatusX
Homework Helper
No. Keep trying, or ask more questions.

An awesome way to think of it is to say,

$$h(x)=\int_{-5}^{\sin x} f(t) dt = F(\sin x) - F(-5)$$

You want d/dx of that, so use the chain rule straight off...

$$\dfrac{d}{dx}(F(\sin x)) = \cos x (f(\sin x))$$

Where $$f(t) = \cos (t^5) + t$$

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