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How do you find the general anti-derivative of #f(x) = 1/2 + (3/4)x^2 - (4/5)x^3#?

Answer 1

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

General anti-derivative can be derived by using applying power rule for integrating f(x) term by term. According to this rule, anti-derivative of #x^n# is #x^n /(n+1)#, where x#!= -1#

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

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

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Answer 2

Adam Adkins

To find the general antiderivative of ( f(x) = \frac{1}{2} + \frac{3}{4}x^2 - \frac{4}{5}x^3 ), integrate each term separately with respect to ( x ):

The antiderivative of ( \frac{1}{2} ) with respect to ( x ) is ( \frac{1}{2}x ).
The antiderivative of ( \frac{3}{4}x^2 ) with respect to ( x ) is ( \frac{1}{4}x^3 ).
The antiderivative of ( -\frac{4}{5}x^3 ) with respect to ( x ) is ( -\frac{1}{5}x^4 ).

Combine these antiderivatives to get the general antiderivative of ( f(x) ):

[ F(x) = \frac{1}{2}x + \frac{1}{4}x^3 - \frac{1}{5}x^4 + 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.