Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

My professor was trying to prove the linearity of the integral. That is,

$\int hf(x)+kg(x) dx=h\int f(x)dx + k \int g(x) dx$, where $f,g$ are continuous functions, and $h,k\in \mathbb{R}$.

The following is the proof he wrote on the board:


Let $\frac{d}{dx}(F(x))=f(x)$ and $\frac{d}{dx}(G(x))=g(x).$

Since differentiation is linear, we have

$\frac{d}{dx}(hF(x)+kG(x))=h\frac{d}{dx}(F(x))+k\frac{d}{dx}(G(x))=hf(x)+kg(x)=\int hf(x)+kg(x) dx=hF(x)+kG(x)=h\int f(x)dx+k\int g(x)dx=h(F(x)+C_1)+k(G(x)+C_2)=hF(x)+hC_1+kG(x)+kC_2=hF(x)+kG(x)+hC_1+kC_2=hF(x)+kG(x)+C$, where $C$ is the constant of integration. QED

I understand most of the proof, however there is one step that makes no sense to me.

It is the following step:

$hf(x)+kg(x)=\int hf(x)+kg(x) dx$

If anyone could explain how this makes sense, I would be grateful.

By the way, it seems like it doesn't make sense to me, as if I tried a concrete example:

If $f(x)=2x$ and $g(x)=3x$, and $h=2$ and $k=3$ then that step is saying that

$2(2x)+3(3x)=\int 2(2x)+3(3x)\implies 4x+9x=\int 4x+9x \implies 13x=\frac{13x^2}{2}$, which doesn't make sense, right?

share|cite|improve this question
Right, it doesn't make sense. I'm trying to figure out what the intention was. But don't hold your breath. – Daniel Fischer Jun 26 '13 at 22:03
I think it's just a handwritten(?) equivalent of a typo, and the incriminated $=$ should have been an implication, $\Rightarrow$. It's still - in my opinion - a terrible and sloppy proof. – Daniel Fischer Jun 26 '13 at 22:08
up vote 0 down vote accepted

I'm guessing it was meant to be something like:


therefore, by integrating both sides it follows that

$\int hf(x)+kg(x) dx=hF(x)+kG(x) +c$


share|cite|improve this answer
Hmm, that's what I was thinking as well. I even asked the professor if what he put on the board was right, and he said yes. – Sujaan Kunalan Jun 26 '13 at 22:19

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.