Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I was wondering what the answer to this question is. I just had a test and I want to make sure I was correct. So I need to find the derivative of $$ \int_{x^6}^{0} \cos(\sqrt{t}) ~ dt $$ (I believe it is $dt$ however I may be wrong and it could have been $dx$.)

The answer I thought was $-3x^2 \cos(x^3)$ but now I realize the answer might be $-6x^5 \cos(x^3)$. Thank you in advance!

share|cite|improve this question
Unfortunately, it is the second answer which is right (the one with $-6x^5$). – André Nicolas Dec 6 '11 at 20:14
Indeed, by the FTC and the chain rule, it's the latter. – The Chaz 2.0 Dec 6 '11 at 20:14
Yeah I had the feeling I did the chain rule wrong thanks though sigh – joe Dec 6 '11 at 20:15
You should post your answer. – David Mitra Dec 6 '11 at 20:33

Here's one way to do this kind of problem (without Wolfram).

Let $F(t)=\int\cos(\sqrt t)\,dt$, so $$F'(t)=\cos(\sqrt t)$$ Then $$\int_{x^6}^0\cos(\sqrt t)\,dt=F(0)-F(x^6)$$ So the derivative is $(F(0))'-(F(x^6))'$. Well, $F(0)$ is a constant, so its derivative is zero, so we just have to figure out $-(F(x^6))'$. By the chain rule, this is $$-F'(x^6)(x^6)'=-\cos(\sqrt{x^6})(6x^5)=-6x^5\cos|x^3|=-6x^5\cos(x^3)$$ since cosine is an even function.

share|cite|improve this answer
"This answer is useful" - especially for those who aren't allowed to access W|A during tests. – The Chaz 2.0 Dec 7 '11 at 3:13

By the Fundamental Theorem of Calculus and the chain rule,

If $F(x) = \int_{a}^{g(x)}f(t)dt$, then $F\ '(x) = f(g(x)) \cdot g'(x)$.

In particular, for $f(t) = \cos(\sqrt{t}); g(x) = x^6$, we have $F \ '(x) = 6x^5 \cdot \cos(x^3)$

(with simplifications as in Gerry Myerson's answer).

But then we have that $\int_a^b \text{(stuff)} = - \int_b^a \text{(stuff)}$, which gives us the "$-$" in front as desired.

share|cite|improve this answer

As you correctly realized the right answer is $-6x^5\cos(x^3)$.

However, just for confirming such results you don't need to ask us, there are efficient means to help yourself:

Note that the latter can already be achieved by just pressing the result in the first link.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.