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I'm stuck with the problem

If $ F(x)=\int_0^{x^3} \sin t^2 dt$ find $F'(x)$

Now, if the upper interval were $x$, the answer would be $\sin t^2$ (right?). However, the upper interval is $x^3$.

I've thought of just working through the integral and then taking the derivative of the answer, but I don't have a clue how to integrate $\sin t^2$ (not $\sin^2t$). I'm not sure if this is even integrable. If it isn't, is the problem solvable at all?

We can't just put $x^3$ in for $t$, so I'm out of ideas of how to solve this problem.

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2 Answers

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In general, $$\frac{d}{dx}\int_{u(x)}^{v(x)}f(t)dt=f(v(x))v^\prime(x)-f(u(x))u^\prime(x).$$ So, in your case, letting $f(t)=\sin(t^2),$ $$\begin{align}F^\prime(x)&=f(x^3)(x^3)^\prime-f(0)\cdot 0\\&=\sin\{(x^3)^2\}\cdot 3x^2-0\\&=3x^2\sin(x^6).\end{align}$$

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Let $G(t)$ be an antiderivative of $\sin(t^2)$

Then $$\frac{d}{dx} \int_0^{x^3} \sin(t^2) dt = \frac{d}{dx} G(x^3)-G(0) = 3x^2G'(x^3) = 3x^2\sin(x^6)$$

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