I'm having trouble with an integral problem which goes like this: Differentiate $$\int_1^{x^3}\arcsin(t)dt$$

The rule I know would be that you make $t$ equal to $x^3$ and then use the chain rule to achieve: $$ 3x^2\arcsin(x^3)$$

But the answer says that it is actually: $$ 3x^2\arcsin(3x^3)$$

Why is this?

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    $\begingroup$ You are totally correct ! One more typo in a textbook. $\endgroup$ – Claude Leibovici Jun 21 '15 at 7:37
  • $\begingroup$ Brilliant! Thank you Claude, this makes me a happy man :) $\endgroup$ – Dave Jun 21 '15 at 7:42
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    $\begingroup$ You are brilliant ! I did not do anything. Cheers :-) $\endgroup$ – Claude Leibovici Jun 21 '15 at 7:44

You are indeed correct. It is a general fact that

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

due to both the FTC and chain rule. Your book likely had a typo.


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