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

Answer 1

#2/3x^3-4x^2+2x+C# for any constant #C#

Note that the antiderivative of #f(x)=p(x)+q(x)+r(x)# is the antiderivative of #p(x)# plus the antiderivative of #q(x)# plus the antiderivative of #r(x)# plus a constant #color(red)(C)#

The derivative of #ax^b# is #b * a x^(b-1)# From which we can see that: The antiderivative of #cx^d# is #c/(d+1) * x^(d+1)#

Using this general form, we have #color(white)("XXX")#antiderivative of #2x^2# is #color(red)(2/3 * x^3)#

#color(white)("XXX")#antiderivative of #-8x (=-8x^1)# is #(-8/2)x^(1+1)=color(red)(-4x^2)#

#color(white)("XXX")#antiderivative of #2 = 2 * x^0# is #(2/1) x^(0+1) = color(red)(2x)#

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Answer 2

Isaiah Allen

To find the antiderivative of ( f(x) = 2x^2 - 8x + 2 ), integrate each term separately using the power rule for integration. The antiderivative is:

[ \int f(x) , dx = \frac{2}{3}x^3 - 4x^2 + 2x + C ]

where ( C ) is the constant of integration.

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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.