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I am revising the syllabus of mathematics and I think I have done enough properties of definite integration. I am listing all the properties that I have studied:

  1. $\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{r=1}^{n}f(\frac{r}{n})=\int_{0}^{1}f(x)dx$
  2. $\left |\int_{a}^{b}f(x)dx\right |\leq\int_{a}^{b}\left | f(x)dx \right |$
  3. $\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$
  4. $\int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx$ if the function $f(x)$ is even. If the function is odd, the the integral simply equals zero.
  5. If $m$ and $M$ are smallest and largest Values of function in the interval $[a,b]$, then the following holds: $m(b-a)\leq\int_{a}^{b}f(x)dx\leq M(b-a)$

The above five properties I have learnt for the examination, now I am wondering if there are more important properties that I am missing here. If yes, could you please explain me. It will be very encouraging for me. Thanks in advance.

Masacroso
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userNoOne
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    What properties are important depends entirely on whoever is writing your exam, and the details of the math course they are writing the exam for. There isn't much we can do to help you, unfortunately. – Arthur Jan 11 '18 at 14:46
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    I don't know if these are on your test, but there's also Integration by parts, u-substitution, Leibniz' rule, etc. – John Lou Jan 11 '18 at 14:53

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You could, for instance remember that:

  1. If $f$ is continuous $$\int_a^b|f(x)|dx=0\Leftrightarrow f(x)=0$$
  2. If $f$ is continuous $$\int_a^bf(x)dx>0\Rightarrow\exists x_0\in[a,b]\epsilon>0:f(x)>0\forall x\in[a,b]\cap(x_0-\epsilon,x_0+\epsilon)$$
  3. If $\lambda\in\mathbb{R}$ then $$\int_a^b(f(x)+\lambda g(x))dx=\int_a^bf(x)+\lambda\int_a^bg(x)dx$$
  4. The Riemann-Integral is not multiplicative.
  5. Integration by parts - as a techinque: $$\int_a^bf(x)g'(x)dx=\left.f(x)g(x)\right|_a^b-\int_a^bf'(x)g(x)dx$$

And, as stated in the comments section, it deepends on how the exam is designed and/or how the class during the semester has been etc.

Vassilis Markos
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