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How do I find the antiderivative of #f(x) = (5x)/(10(x^2-1))#?

Answer 1

I would first manipulate the argument to get it in a form which is easier to integrate: Simplifying the #5# and #10# and transforming #x^2-1# in a product you get:

#int(5x)/(10(x^2-1))dx=int(x)/(2(x-1)(x+1))dx=#

I then rearrange to get a sum introducing an additional #1/2#;

#=int1/4*(1/(x-1)+1/(x+1))dx=#

(which is equivalente to the starting one: #int(x)/(2(x-1)(x+1))dx#)

And finally:

#=int1/4*(1/(x-1)+1/(x+1))dx=1/4[ln(x-1)+ln(x+1)]+c# or #=1/4[ln(x^2-1)]+c#

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Answer 2

Isabella Ackerman

To find the antiderivative of ( f(x) = \frac{5x}{10(x^2-1)} ), first, rewrite the function by factoring out common terms from the denominator. Then, use algebraic manipulation and apply integration techniques to find the antiderivative. In this case, partial fraction decomposition and integration by substitution can be employed. After integrating, you'll obtain the antiderivative.

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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.