B Calculate the expression of the antiderivative

[hugo_faurand]

Hello everyone !
I've started to work on integral and I wonder if it's possible to calculate the expression of the antiderivative with the expression of the "integrand"¹ rather than use a table with the function and its antiderivative.

Thank you in advance !

¹( I'm french and I d'ont know the translation for this word, so I make an assumption and I put quotes)

 

[fresh_42]

Hello everyone !
I've started to work on integral and I wonder if it's possible to calculate the expression of the antiderivative with the expression of the "integrand"¹ rather than use a table with the function and its antiderivative.

Thank you in advance !

¹( I'm french and I d'ont know the translation for this word, so I make an assumption and I put quotes)

On can write the antiderivative as a limit of Riemannian sums, which are expressions of the integrand¹, but this doesn't give you a closed expression. This can only eventually be done after you know the result, as for $\int \sin x dx = -\cos x = \sin (x-\frac{\pi}{2})$ which is more by chance than by an actual dependency.

¹) Neither am I, so this might still be the wrong word.

 

[DrClaude]

http://mathworld.wolfram.com/Integrand.html

The quantity being integrated, also called the integral kernel.

 

[BvU]

Yes. If $\ {dF(x)\over dx} = f(x)\$ then $F(x) - F(0) = \int_0^x f(u) du \$, and $F(x)$ is an antiderivative of $f(x)$. But is this what you meant to ask ?

 

[hugo_faurand]

Yes. If $\ {dF(x)\over dx} = f(x)\$ then $F(x) - F(0) = \int_0^x f(u) du \$, and $F(x)$ is an antiderivative of $f(x)$. But is this what you meant to ask ?

In fact, if I have the expression of the integrand, Can I calculate the antiderivative ?
For example, I search the antiderivative of x². I would like to know if it exists a kind of formula to calculate the antiderivative with the expression of the integrand.

 

[BvU]

You know that $\ {d\over dx} x^n = nx^{n-1},\$ so if $\ F(x) = {1\over n+1}x^{n+1},\$ then F(x) is an antiderivative of $x^n$.

This satisfies the criterion 'a formula to calculate the antiderivative' for a specific kind of function $f$.
A general recipe is not available, so it remains a kind of 'metier', or better: 'artisanat'

 

[fresh_42]

In fact, if I have the expression of the integrand, Can I calculate the antiderivative ?
For example, I search the antiderivative of x². I would like to know if it exists a kind of formula to calculate the antiderivative with the expression of the integrand.

In this case, it is the formula $x^n \longmapsto \dfrac{1}{n+1}x^{n+1}$ but this only covers polynomials, and in general, the answer is no. You cannot write down straight away a formula for, say $\int{\dfrac{\cot (1+x^2)}{\tan (1-x^2)}\, dx}$.

 

[hugo_faurand]

So, I just have to learn by heart my derivative.
Another little question can I use this expression to calculate all the derivatives I want or Is there exeptions ?

$$ f'(x) = \lim_{dx\to 0} \frac{f(x+dx)-f(x)}{dx}$$

 

[BvU]

Do not forget to take the limit $dx\rightarrow 0$ !
If the limit does not exist, then the derivative also does not exist !

 

[hugo_faurand]

Do not forget to take the limit $dx\rightarrow 0$ !
If the limit does not exist, then the derivative also does not exist !

Yes, corrected !