# At what value does this integral reach its minimum?

The problem is to find the value of $$a>1$$ at which this integral $$\int_{a}^{a^2} \frac{1}{x}\ln\Big(\frac{x-1}{32}\Big)dx$$

reaches its minimum value. I don't want to know the number, I just want feedback on the ideas I'm trying. Considering that $$f(x)=\frac{1}{x}\ln\Big(\frac{x-1}{32}\Big)<0$$

whenever $$x<33$$ and the fact that $$f(x)=0$$ at $$x_0=33$$, we have that all the area associated with the graph of this function is negative until the point $$x_0$$. So we want to find the value of $$a$$ which yields the greatest portion of this "negative area". The fundamental theorem of calculus states that $$\int_a^bf(x) = F(b) - F(a)$$

which gives us an expression which can be differentiated. Particularly, the first derivative is $$f' =\frac{1}{a^2}\ln\Big(\frac{a^2-1}{32}\Big)-\frac{1}{a}\ln\Big(\frac{a-1}{32}\Big)$$

or $$f'=\frac{1}{a^2}\ln\Big(\frac{a^2-1}{32}\Big)\cdot 2a-\frac{1}{a}\ln\Big(\frac{a-1}{32}\Big)\cdot 1$$

I think the second is correct since, per the fundamental theorem, we are applying the primitive $$F$$ to the upper and lower bound functions $$a^2$$, and $$a$$ - so this is a sum of two composite functions differentiated by the chain rule. It depends on the correct interpretation of $$a$$ I think.

If we differentiate again, we can find the values of $$a$$ for which $$f(a)''>0$$. This will be value of $$a$$ where the integral reaches its minimum value. However, applying the second derivative test to an integral doesn't seem proper since it is a tool for studying the concavity of $$f$$ - what would be the interpretation here?

Your second expression for $$f'(a)$$ is correct. Also, you shouldn't differentiate it again, but rather solve for the critical points of $$f'(a)$$ to find the minimum value of the integral because the integral is $$f(a)$$: $$f'(a)=0=\frac{1}{a}\left(2\ln{\left(a^2-1\right)}-2\ln{32}-\ln{\left(a-1\right)}+\ln{32}\right)$$ $$\ln{32}=\ln{\left(a-1\right)}+2\ln{\left(a+1\right)}$$
Now, solve for $$a$$ then test if its a minimum or maximum by using the first derivative test.