Thread: How To Find The Antiderivative

I have been asking this question, but I never seem to get a direct answer.... Is there any type of formula to find an antiderivative?

it depends , but alot of the time when you have a variable with a power, you just add 1 to the power and divide all of that by what you get for instance :

$\displaystyle \int x^{2}$

becomes

$\displaystyle \frac{x^{2+1}}{3}$

$\displaystyle \frac{x^{3}}{3}$

Thank you... that helps alot... but what about if it's not to a power say 2/X^2

finding an anti derivative is based on these steps:
1) Known formula,which gives an immediate integral
now,when things get serious,you have to use at least one of those:
2)U substitution
3)Integration by parts
4)Partial Fractions
5)Trig substitution
6)complete the square

that`s about it i think

You have to see it as $\displaystyle 2x^{-2}$

if you integrate, your 2 goes in front because it's a constant and you just have to apply the trick :

$\displaystyle 2\int x^{-2}$

which is

$\displaystyle 2(\frac {x^{-1}}{-1})$

=

$\displaystyle \frac{-2}{x}$

Originally Posted by katchat64

Thank you... that helps alot... but what about if it's not to a power say 2/X^2

but it is "to a power" ...

$\displaystyle \frac{2}{x^2} = 2x^{-2}$

antiderivative is $\displaystyle -2x^{-1} = -\frac{2}{x}$

there's no formula, but to find the integral

just add the exponent by 1, and then divide the part you integrated by your new exponent

Ok then... how would you take the antiderivative of

$\displaystyle \int_a^b{3^x\,dx}$

Originally Posted by katchat64

I have been asking this question, but I never seem to get a direct answer.... Is there any type of formula to find an antiderivative?

There are many different formulae (the better word is techniques) .... it all depends on what is being anti-differentiated.

Originally Posted by katchat64

Ok then... how would you take the antiderivative of

$\displaystyle \int_a^b{3^x\,dx}$

you have to memorize one of those rules

the rule for 3^x would be ln3 x 3^x

dx/dy[a^x]=(lna)a^x

so then use the fundamental calculus theorem and plug it in [(ln3)3^x] from b to a