SOLUTION: Find the most general antiderivative of each function. a.) f(x)=20x^3+12x^2+4 b.) g(x)=3x^4-x^3+6x^2/x^4 c.) h(t)=e^x+sinx+14(5root(x^2))

Algebra ->  Equations -> SOLUTION: Find the most general antiderivative of each function. a.) f(x)=20x^3+12x^2+4 b.) g(x)=3x^4-x^3+6x^2/x^4 c.) h(t)=e^x+sinx+14(5root(x^2))      Log On


   

Question 819248: Find the most general antiderivative of each function.
a.) f(x)=20x^3+12x^2+4
b.) g(x)=3x^4-x^3+6x^2/x^4
c.) h(t)=e^x+sinx+14(5root(x^2))
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The phrase "most general" must mean that you need to add a constant.

a.) The derivative of 5x%5E4%2B4x%5E3%2B4x is f%28x%29=20x%5E3%2B12x%5E2%2B4 ,
but the derivative of highlight%285x%5E4%2B4x%5E3%2B4x%2BK%29 where the constant K represents any number is f%28x%29 too.

b.) You wrote g(x)=3x^4-x^3+6x^2/x^4=3x%5E4-x%5E3%2B6x%5E2%2Fx%5E4=3x%5E4-x%5E3%2B6x%5E%28-2%29
The antiderivative of that would be
3x%5E5%2F5-x%5E4%2F4%2B6%28x%5E%28-1%29%2F%28-1%29%29%2BK=3x%5E5%2F5-x%5E4%2F4%2BK-6%2Fx
However, if you meant g'(x)=(3x^4-x^3+6x^2)/x^4=%283x%5E4-x%5E3%2B6x%5E2%29%2Fx%5E4=3-x%5E%28-1%29%2B6x%5E%28-2%29=3x-1%2Fx%2B6%2Fx%5E2 ,
the antiderivative is
3x-ln%28x%29%2B6%28x%5E%28-1%29%2F%28-1%29%29%2BK=3x-ln%28x%29-6%2Fx%2BK

c.) If you meant h(x)=e%5Ex%2Bsin%28x%29%2B14root%285%2Cx%5E2%29=e%5Ex%2Bsin%28x%29%2B14x%5E%222%2F5%22,
The antiderivative of that function can be calculated as