# What are definite integrals that use functions in one or both of its limits?

I've seen these types of questions before, but I think I missed a formal explanation of them. I have a solution to the question in front of me, so my question is not related to what the answer is - but what is the name for this type of problem? I don't quite understand why we're given these kinds of problems other than just because...

Find and simplify the derivative of $\displaystyle\int_{2/x}^{1}\frac{t^{3}}{1+t^{5}}\ dt$

\begin{align} f(x)&=\int_{2/x}^{1}\frac{t^{3}}{1+t^{5}}\ dt\\ &=-\int_1^{2/x}\frac{t^{3}}{1+t^{5}}\ dt\\ f'(x)&=\frac{(2/x)^3}{1+(2/x)^5}\cdot \left(-\frac{2}{x^2}\right)\\ &=\frac{16}{x^5+32} \end{align}

I'm trying to prepare for my final exam on Wednesday and I'd like to get a bit more background on this type of question so I can better understand what I'm doing if I am asked.

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I'm not sure if this is what you're after, but what you're using is the fundamental theorem of calculus. – Git Gud Apr 14 '13 at 15:37
@GitGud I know what the FTC is... and I realize that it is being used in this process... but I'm curious about the type of problem where you're given a definite integral with variables in one or both of its limits. – agent154 Apr 14 '13 at 15:40
I really don't understand what you're looking for. Hopefully someone else will. – Git Gud Apr 14 '13 at 15:42
Check out Liebnitz rule – Gautam Shenoy Apr 14 '13 at 15:45

Let $f(t)$ be any integrable function and define $g(x)=\int_{a(x)}^{b(x)}f(t)dt$. Then, by the fundamental theorem of calculus, $g(x)=F(b(x))-F(a(x))$ where $F(x)$ is a function with $F'(t)=f(t)$. Now apply the chain rule: $$g'(x)=F'(b(x))\cdot b'(x)-F'(a(x))\cdot a'(x)=f(b(x))\cdot b'(x)-f(a(x))\cdot a'(x)$$ In your case, $b(x)=1$, so $b'(x)=0$ and $a(x)=2/x$ hence $a'(x)=-2/x^2$