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How do you integrate #f(x)=x^3-3x-2x^-4# using the power rule?

Answer 1

William Abdalla

#1/4x^4-3/2x^2+2/3x^-3+C#

for polynomials

#intx^ndx=x^(n+1)/(n+1)+C,n!=-1#

#int(x^3-3x-2x^(-4))dx#

#=x^(3+1)/(3+1)-3x^(1+1)/(1+1)-2x^(-4+1)/(-4+1)+C#

#=x^4/4-(3x^2)/2-(2x^-3)/-3+C#

tidying up

#1/4x^4-3/2x^2+2/3x^-3+C#

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Answer 2

Charles Arocha

To integrate the function (f(x) = x^3 - 3x - 2x^{-4}) using the power rule, you first find the antiderivative of each term separately, then sum them up.

The antiderivative of (x^3) is (\frac{x^4}{4}), the antiderivative of (-3x) is (-\frac{3x^2}{2}), and the antiderivative of (-2x^{-4}) is (\frac{2x^{-3}}{-3}).

Therefore, integrating (f(x)) yields (\frac{x^4}{4} - \frac{3x^2}{2} + \frac{2x^{-3}}{-3} + 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.