Need Help with Solving an Integral Problem? Let Our Experts Guide You!

[Waffle07]

Ok I can't 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.

 

[StatusX]

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

 

[Waffle07]

Its says the derivative of g(x) is equal f(x), I know the Fundamental Theorem of Calculus, but I just don't know how to use it alongside the chain rule. Our book only has one example of it, and it doesn't 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 won't take it.

 

[StatusX]

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]

Its says the derivative of g(x) is equal f(x)

No matter what f and g are? What a remarkable theorem!

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.

 

[Waffle07]

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

 

[StatusX]

No. Keep trying, or ask more questions.

 

[Waffle07]

K the answer is cosx*cos(sin(x)^5)+cosx*sinx

 

[Pseudo Statistic]

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

 

[Georgie Cartier]

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