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How do you integrate #f(x)=(x^3+5x)/((x^2+4)(x-3)(x+9))# using partial fractions?

Answer 1

Ezra Adams

See answer below:

Continuation:

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Answer 2

Ezekiel Alexander

To integrate ( f(x) = \frac{x^3 + 5x}{(x^2 + 4)(x - 3)(x + 9)} ) using partial fractions, follow these steps:

Factor the denominator: [ (x^2 + 4)(x - 3)(x + 9) ]
Write ( f(x) ) as a sum of partial fractions: [ f(x) = \frac{A}{x^2 + 4} + \frac{B}{x - 3} + \frac{C}{x + 9} ]
Clear the denominators by multiplying both sides of the equation by ( (x^2 + 4)(x - 3)(x + 9) ).
Expand and simplify the equation to solve for ( A ), ( B ), and ( C ).
After finding the values of ( A ), ( B ), and ( C ), rewrite ( f(x) ) in terms of those values.
Integrate each term separately using standard integration techniques.
Sum up the integrated terms to get the final result.

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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.