Home > Calculus > Integrals of Polynomial functions

What is the antiderivative of #F(x)=9x^2-8x+8#?

Answer 1

#3x^3-4x^2+8x+C#

The integral #int x^(n)\ dx=(x^(n+1))/(n+1)+C# when #n!=-1# should be memorized (it's the reverse of the Power Rule #d/dx(x^(n))=nx^(n-1)#).

Also, up to addition of a constant of integration #C#, indefinite integrals satisfy the linearity property:

#int (a_{1}f_{1}(x)+a_{2}f_{2}(x)+\cdots+a_{n}f_{n}(x))\ dx#

#=a_{1}int f_{1}(x)\ dx+a_{2}int f_{2}(x)\ dx+\cdots+a_{n}int f_{n}(x)\ dx#

This means that we integrate term-by-term and "carry the constants along for the ride", so to speak. Then, tack on the #+C# at the end.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

William Abdalla

The antiderivative of F(x) = 9x^2 - 8x + 8 is F(x) = 3x^3 - 4x^2 + 8x + C, where C is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.