How to approach this calculus question.

Question

I was trying to look in different calculus questions in one textbook, and I found this one as interesting but pretty hard for me. I have only done high school calculus, so I don't know any tricks on how to approach this question or similar ones.

I tried integrating in couple different ways, however none of those worked. I'm not sure how to take derivative of an indefinite integral, since it seems not that straightforward.

$$\frac{d}{dx} \int_x^{0}\frac{t}{cost}dt$$

Answer 1

It happens that $\int_a^bf(t)\,\mathrm dt=-\int_b^af(t)\,\mathrm dt$. So$$\frac{\mathrm d}{\mathrm dx}\int_x^0\frac t{\cos t}\,\mathrm dt=-\frac{\mathrm d}{\mathrm dx}\int_0^x\frac t{\cos t}\,\mathrm dt.$$And, by the Fundamental Theorem of Calculus,$$\frac{\mathrm d}{\mathrm dx}\int_0^x\frac t{\cos t}\,\mathrm dt=\frac x{\cos x}.$$

Answer 2

$$\frac{d}{dx} \int_x^{0}\frac{t}{cost}dt = - \frac{d}{dx} \int_0^{x}\frac{t}{cost}dt$$

Now use the Fundamental Theorem of Calculus.

Note that derivative of an anti-derivative is the original function.