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

Question

Paisley Johnson · Accepted Answer

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

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|))#

Paisley Johnson · Answer

Like this :

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)#

Christopher Agresta · Answer

undefined To find the antiderivative of $ f(x) = 8x^3 + 5x^2 - 9x + 3 $, you can use the power rule for integration. The power rule states that the antiderivative of $ x^n $ with respect to $ x $ is $ \frac{x^{n+1}}{n+1} + C $, where $ C $ is the constant of integration.

Applying the power rule to each term of $ f(x) $:

- The antiderivative of $ 8x^3 $ is $ \frac{8}{4}x^4 = 2x^4 $.
- The antiderivative of $ 5x^2 $ is $ \frac{5}{3}x^3 = \frac{5x^3}{3} $.
- The antiderivative of $ -9x $ is $ -\frac{9}{2}x^2 = -\frac{9x^2}{2} $.
- The antiderivative of $ 3 $ is $ 3x $.

So, the antiderivative of $ f(x) $ is $ 2x^4 + \frac{5x^3}{3} - \frac{9x^2}{2} + 3x + C $, where $ C $ is the constant of integration.