Evaluate

F(x) is called the antiderivative of f(x).

F(x)=f(x)\displaystyle F(x)=\int f(x)

So, what is the derivative of f(x)dx\displaystyle \int f(x)dx?.

That is, what is ddx[F(x)]=ddxf(x)dx=\displaystyle \frac{d}{dx}[F(x)]=\frac{d}{dx}\int f(x)dx=

What happens when you differentiate an antiderivative?.


Besides, sin(x5)dx\displaystyle \int sin(x^{5})dx is a nasty integration that is beyond elementary calc.

The point is not to integrate, but use the basic knowledge of what an antiderivative is.
 
Oh, OK then. That is a different matter. In that case, your last post is correct.

Remember, F(x) normally means the ANTI-derivative. As in integration. If you are not there yet, you will learn about it later.
 
CatchThis2 said:
The integral of sin(x^5)dx <<<< Where did this come from?

The f '(x)= (cosx^5) (5x^4) Correct ?

If

f(x) = sin (x[sup:3lujo3hg]5[/sup:3lujo3hg])

then, using chain rule

f'(x) = cos(x[sup:3lujo3hg]5[/sup:3lujo3hg]) * 5 * x[sup:3lujo3hg]4[/sup:3lujo3hg] = 5 * x[sup:3lujo3hg]4[/sup:3lujo3hg] * cos(x[sup:3lujo3hg]5[/sup:3lujo3hg])
 
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