# Intergral ?

Find the derivative.

d/dx $$\int$$$$\stackrel{x^5}{o}$$ sint dt

I came up with sin x^5.

The answer is 5x^4 sin (x^5).

Not sure what I'm missing here.

Last edited:

Zryn
Gold Member
Is the question:

Integrate x^5*sin(t) with respect to 't', then differentiate the result with respect to 'x'?

x^5 and 0 are the limits, right?

First: you integrate sint with respect to t.
Second: Apply the limits to the value you obtained for the integer. You'll get two terms, get it?

Then, derive this result with respect to x.

LCKurtz
Homework Helper
Gold Member
x^5 and 0 are the limits, right?

First: you integrate sint with respect to t.
Second: Apply the limits to the value you obtained for the integer. You'll get two terms, get it?

Then, derive this result with respect to x.

That works in this problem but the point is likely to use Leibnitz rule for differentiating an integral with variable upper limit. It would work even if you had something more difficult than sin(t) that you couldn't find the antiderivative for. The basic Leibnitz rule is:

$$\frac d {dx}\int_a^{g(x)} f(t)\, dt = f(g(x))\cdot g'(x)$$

Mark44
Mentor
The repaired LaTeX is below.
Find the derivative.

d/dx $$\int$$$$\stackrel{x^5}{o}$$ sint dt

I came up with sin x^5.

The answer is 5x^4 sin (x^5).

Not sure what I'm missing here.

$$d/dx \int_0^{x^5} sint dt$$
To the OP: click the integral to see the LaTeX code I used.
The integral in the original post was confusing to at least one person who didn't understand that x5 was one of the limits of integration.

Also, don't use o (the letter) where 0 (the numeral) is intended.

BTW, there is no such word as "intergral" - the word is integral.