# How do you integrate #int (3x^2 + 10x -5)/ ( (x+1)^2(x-2) )# using partial fractions?

First things first:

Thus, we can now say:

This system provides us with:

Now let's discuss the integral:

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To integrate the expression (\frac{{3x^2 + 10x -5}}{{(x+1)^2(x-2)}}) using partial fractions, first express the fraction as a sum of simpler fractions.

Perform long division if necessary to ensure the numerator's degree is less than the denominator's degree. Then, express the fraction in the form:

[\frac{{3x^2 + 10x -5}}{{(x+1)^2(x-2)}} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x-2}]

Find the values of A, B, and C by multiplying through by the common denominator and comparing coefficients.

After finding the values of A, B, and C, integrate each term separately.

Finally, combine the integrals and simplify if necessary.

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To integrate the given rational function (\frac{3x^2 + 10x - 5}{(x+1)^2(x-2)}) using partial fractions, we first need to express it as a sum of simpler fractions. The denominator has repeated linear factors, so the partial fraction decomposition will have the form:

[\frac{3x^2 + 10x - 5}{(x+1)^2(x-2)} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x-2}]

Multiplying both sides by ((x+1)^2(x-2)) to clear the denominators, we get:

[3x^2 + 10x - 5 = A(x+1)(x-2) + B(x-2) + C(x+1)^2]

Expanding the right side and collecting like terms, we get:

[3x^2 + 10x - 5 = A(x^2 - x - 2) + B(x-2) + C(x^2 + 2x + 1)]

[3x^2 + 10x - 5 = (A + C)x^2 + (-A + B + 2C)x + (-2A - 2B + C)]

Comparing coefficients of (x^2), (x), and the constant term, we get the following system of equations:

[A + C = 3] [-A + B + 2C = 10] [-2A - 2B + C = -5]

Solving this system of equations, we find (A = 1), (B = 5), and (C = 2). Therefore, the partial fraction decomposition of the given expression is:

[\frac{3x^2 + 10x - 5}{(x+1)^2(x-2)} = \frac{1}{x+1} + \frac{5}{(x+1)^2} + \frac{2}{x-2}]

Now, we can integrate each term separately:

[\int \frac{1}{x+1} , dx = \ln|x+1|] [\int \frac{5}{(x+1)^2} , dx = -\frac{5}{x+1}] [\int \frac{2}{x-2} , dx = 2\ln|x-2|]

So, the integral of (\frac{3x^2 + 10x - 5}{(x+1)^2(x-2)}) is:

[\ln|x+1| - \frac{5}{x+1} + 2\ln|x-2| + C]

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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.

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.

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.

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