Problem
let $$ f(x)=\frac{11}{x}+4 \int_{0}^{1}tdt, \hspace{5mm} x> 0 $$
$(a)$ Compute $\int_{1}^{2}f(x)dx$
$(b)$ Define that integral function of $f$ that goes through point $(1,-2)$
Attempt to solve
$(a)$ is straight forward:
$$ f(x) = \frac{11}{x}+2 $$ $$ \int_{1}^{2} \frac{11}{x}+2 dx = \big{/}_{0}^{1} 11\ln|x|+2x + C $$ $$ = (11\cdot \ln|2|+2\cdot 2)-(11\cdot \ln|1| + 2\cdot 1) $$ $$ =11 \cdot \ln|2|+2 $$
Problem is i don't understand what does the $(b)$ ask for ? Integral function of $f$ would be defined as:
$$ \int f(x) dx = \int \left( \frac{11}{x} + 4\int_{0}^{1}tdt dx \right) = \int \frac{11}{x} + 2 dx$$
$$ =11 \cdot \ln|x| + 2x + C $$
Now i think i need to solve the constant in a way that the function satisfies condition $f(1)=-2$
$$F(x)=11 \cdot \ln|x| + 2x + C$$ $$ 11 \cdot \ln|1| + 2 \cdot 1 + C = -2 $$ $$ C = -4 $$
So our integral function has to be
$$ F(x) = 11 \cdot \ln|x| + 2x - 4 $$
Just to make sure
$$ F(1) = 11 \cdot \ln|1| + 2\cdot 1 - 4 = -2 $$
I think i managed to solve it ? I don't deny the possibility of an error but to me this would seem that i at least understood the question ?
Don't know why this keeps happening that the only thing stopping me from solving a problem is that i am thinking i cannot solve it or i don't understand it but up on closer inspection i can solve it and i do understand the question ? I think the initial thought "I cant solve this / this looks hard" is not good.
Anyway i would highly appreciate if someone notices a flaw please let me know.
I think you have solved part (b) as intended.
I don't know your textbook, but I was guessing 'integral function' means 'a function that is the integral of $\text{f(x)}$'. Since there are infinitely many functions which differ by a constant, it would make sense for the question to choose one: by specifying it passes through $(1, -2)$.
I might have an idea on what's stopping you from solving your problem. When I took AP Calc (at least it's now over!), I found that I knew all the material, but still struggled on solving even the first part of the question. What worked for me was to read the question carefully, then choose a method. When I had a clear aim on how to solve the question, then everything was simpler and faster, especially during the working-out.
Hope this helps.