So I recently came across the formula $$\int_b^a [f(x)+f^{-1}(x)]dx=a^2-b^2$$ which apparently came up in a final of the MIT integration bee, and is true when $a,b$ are fixed points of $f$. I was wondering how one would go about proving this formula? I have no clue how to even make a start!
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If $a,b$ are fixed points of $f$, then $$ \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $$   In the words of 2014 MIT Integration Bee Champion (Carl Lian), the above property was responsible for the champion's victory in the 2013 MIT Integration Bee. How does one go about proving this property?
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Find the integral solution of $$a^b -  b^a = 3$$ I am a student of class 10 and got this question in a maths competetive exam.  I have tried converting into a log equation in one variable, or solve it by using parity but couldn't solve it.  Thanks in advance
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Assuming $a&lt;b$, function f(x) is continous at [a,b], and we have  $\int_a^bf(x)dx=\int_a^bxf(x)dx=\int_a^bx^2f(x)dx=0$  Prove that $\exists \ x_1,x_2,x_3\text {(different from each other)}  \in(a,b)$ satisfying $f(x_1)=f(x_2)=f(x_3)=0$.  I have proved the existence of one zero point by differential mean value theorem, but have no idea about going on.  Thanks for any solution or hint.
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Let $f$ be a real-valued function on $A$ in $E^n$. Show that is $\int_A f$ exists, then so does $\int_A |f|$ and $\left | \int_A f\right | \le \int_A |f|$.   I am trying to finish Rosenlicht's Introduction to Analysis and the last chapter is awful. But I do want to finish it because I have come this far. Could someone show me how to do this, so I can at least have some example problems to understand the material. This chapter (Multiple Integrals) really just left more confused than before I started reading the chapter.  Thank you in advance.
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Is it true that  $$\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$$  Just making sure.   If not, how about:  $$\int_a^b f(x) dx = (f(b)-f(a))b - \int_{f(a)}^{f(b)}f^{-1}(x)dx$$   I'm having a hard time concentrating right now, and I'm trying to figure out how to get the area under a curve when the function is inverted.
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when I apply $f^{-1}$ to both sides $f(f(x))=f(x)$ $(1)$ we get $f(x)=x$ $(2)$.  are $1$ and $2$ equivalent?
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Integrate $\dfrac{dx}{(a^2 \cos^2x+b^2 \sin^2x)^2}$.  I can go up to the residue formula like in this example   but then I just can't end up with the result for when $n=2$. I keep messing up my math. Can someone please show me how to get to the answer when $n=2$? I think I've spent 3+ hours trying to get to it and no luck unfortunately. Thank you!
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$\int_{-1}^{1}(1-x^2)^ndx$  I have trouble with finding recurrence formula for this integral. $n$ is natural parameter. I've tried to split up $(1-x^2)^n = (1+x)^n(1-x)^n$ and then to integrate partially, but it only makes things more complicate.  Maybe substitution $x=sint$ can lead to solution? When I apply it I get:   $\int_{-\pi/2}^{\pi/2}(cost)^{n+1}dt$  What to do next then?
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Prove that if $f:[a,b] \rightarrow \mathbb{R}$ is integrable, then so is $|f|$ and $\left|\int_{a}^{b}f(x)dx\right| \leq \int_{a}^{b}|f(x)|dx$   I already proved that |f| is integrable, but how do I show that $|\int_{a}^{b}f(x)dx| \leq \int_{a}^{b}|f(x)|dx$   ?
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Let $f\in C ((-\infty,+\infty)).  $ If $ \forall  a,b\in(-\infty,+\infty),\int_{a}^{b}f^{2}(x)dx \leq  f(a)+f(b), $ then$f(x)\equiv 0$.   I can prove $f(x)\geq 0,$ so I want use the reduction to absurdity in the next proof that may refuse $f(x)&gt;0.$ In my way of thinking ,I tried but it is seemingly difficult  to deny $f(x)&gt;0.$ Could you please give me some hints?
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Find the exact value of     $$ \int_{0}^{a} \frac{1}{\sqrt {a^2-x^2}} \mathrm {dx} $$   Where, $a$ is a positive constant  Hi, guys can give me tips to solve this ? Should we use like u substitution?
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If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive, so $$\int_{0}^{1} \frac{f(x)}{f(x)+f(1-x)}dx=\frac{1}{2}$$???  all examples that I tested have worked.
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If $f \colon [a,b]\rightarrow \mathbb{R}$, then show that there exists $c \in [a,b]$ such that $c \in [a,b]$ such that $\int_a^c f = \int_c^b f$.  I think that I have to proceed with the Mean Value Theorem of Integrals...
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Could someone advise me on how to approach this problem:  without  the use of argument principle ?    Here is my attempt: Suppose $f(z)$ has two zeroes at $z=a.$ Let $f(z) = \sum^{\infty}_{n=0}a_n(z-a)^n, \forall z. \ $   Then $f(z)=(z-a)^2 \left(\dfrac{a_{0}}{(z-a)^2} +\dfrac{a_1}{z-a}+a_2+a_3(z-a)+...\right)$   $\implies a_0=a_1=0.$  $\implies f(z)= (z-a)^2\left(a_2+a_3(z-a)+a_4(z-a)^2+...\right)$  $\implies .... ?$  Thank you.
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Let $f:X\to X $ be continuous. Show that if $X=[0,1] $, $f $ has a   fixed point(i.e. there exists  $x$ such that $f (x)=x$). What happens   if $X $ equals  $[0,1) $ or $(0,1) $?   First part of the question is an immediate consequence of intermediate value theorem(for a proof, see ). I think $f (x)=x^2$ is a counterexample when $X=(0,1) $, since $x^2\lt x $ for all $x\in (0,1) $. But to be honest, I don't understand what causes breakdown of the fixed point theorem on $(0,1)$, since IVT only requires connectedness of domain. Is this related to the non-compactness of $(0,1)$?  Also I can't think of any counterexample for the case $X=[0,1)$(assuming fixed point is somehow related to compactness).  Any help is appreciated.   Thank you.
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Suppose that $f$ is a function defined in $[a;b]$ to $[a;b]$ and continuous on $[a;b]$. The problem is I haven't the definition of the function, this is more abstract, but even if how can I prove that $f$ would have a fixed point?
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I have proved that if $f \in R[a,b]$ and given $\epsilon &gt; 0$ there exists a continuous function $g$ such that $$\int_a^b |f-g| &lt; \epsilon$$ I was wondering if using this fact there is some way to show that there is also some continuous function $h$ such that $$\int_a^b |f-h|^2 &lt; \epsilon$$ Any help will be appreciated, thanks :)