How do you find the antiderivative of #f(x)=8x^3+5x^2-9x+3#?

2 Answers
Apr 5, 2018

Like this :

Explanation:

The anti-derivative or primitive function is achieved by integrating the function.

A rule of thumb here is if asked to find the antiderivative/integral of a function which is polynomial:
Take the function and increase all indices of #x# by 1, and then divide each term by their new index of #x#.

Or mathematically:

#int x^n=x^(n+1)/(n+1)(+C)#

You also add a constant to the function, although the constant will be arbitrary in this problem.

Now, using our rule we can find the primitive function, #F(x)#.

#F(x)=((8x^(3+1))/(3+1))+((5x^(2+1))/(2+1))+((-9x^(1+1))/(1+1))+((3x^(0+1))/(0+1))(+C)#

If the term in question does not include an x, it will have an x in the primitive function because:

#x^0=1# So raising the index of all #x# terms turns #x^0# to #x^1# which is equal to #x#.

So , simplified the antiderivative becomes:

#F(x)=2x^4+((5x^3)/3)-((9x^2)/2)+3x(+C)#

Apr 5, 2018

#2x^4+5/3x^3-9/2x^2+3x+C#

Explanation:

The anti-derivative of a function #f(x)# is given by #F(x)#, where #F(x)=intf(x) \ dx#. You can think of the anti-derivative as the integral of the function.

Therefore,

#F(x)=intf(x) \ dx#

#=int8x^3+5x^2-9x+3#

We are going to need some integral rules to solve this problem. They are:

#inta^x \ dx=(a^(x+1))/(x+1)+C#

#inta \ dx=ax+C#

#int(f(x)+g(x)) \ dx=intf(x) \ dx+intg(x) \ dx#

And so, we get:

#color(blue)(=barul(|2x^4+5/3x^3-9/2x^2+3x+C|))#