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How do you find the antiderivative #f(x)=6/x^5#?

Answer 1

Aurora Alvarado

Note that #\int x^{n}\ dx=x^{n+1}/(n+1)+C# when #n!=-1# and that #\int af(x)\ dx=a\int f(x)\ dx# for any constant #a#.

Hence #\int 6/(x^5)\ dx=6\int x^{-5}\ dx=6* x^{-4}/(-4)+C=-3/(2x^4)+C#

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Answer 2

Scarlett Allison

The antiderivative of (f(x) = \frac{6}{x^5}) can be found using the power rule for integration. The power rule states that the antiderivative of (x^n) is (\frac{x^{n+1}}{n+1}), where (n \neq -1).

Applying the power rule to (f(x) = \frac{6}{x^5}), we get:

[ \int \frac{6}{x^5} , dx = 6 \int x^{-5} , dx = 6 \cdot \frac{x^{-5+1}}{-5+1} + C = -\frac{6}{4x^4} + C = -\frac{3}{2x^4} + C ]

Therefore, the antiderivative of (f(x) = \frac{6}{x^5}) is (-\frac{3}{2x^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.