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How do I find the integral #int(5x^2+3x-2)/(x^3+2x^2)dx# ?

Answer 1

Samantha Adkison

#int{5x^2+3x-2}/{x^3+2x^2} dx=2ln|x|+1/x+3ln|x+2|+C#

Let us look at some details.

#int{5x^2+3x-2}/{x^3+2x^2} dx#

by Partial Fraction Decomposition (details not included),

#=int (2/x-1/x^2+3/{x+2}) dx#

by Log Rule and Power Rule,

#=2ln|x|+1/x+3ln|x+2|+C#

I hope that this was helpful.

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Answer 2

Evelyn Agard

To find the integral ∫(5x^2 + 3x - 2)/(x^3 + 2x^2) dx, you can use partial fraction decomposition. First, factor the denominator x^3 + 2x^2 as x^2(x + 2). Then, express the fraction as the sum of two fractions with undetermined coefficients: A/x + B/x^2 + C/(x + 2). Solve for A, B, and C by equating coefficients. Once you have A, B, and C, integrate each term separately, and then sum them up. The result will be the integral of the original expression.

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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.