# How do you integrate #(6x^2+1)/(x^2(x-1)^2)# using partial fractions?

The integral is

First, let's figure out the partial fractions' coefficients.

So

#int((6x^2+1)dx)/(x^2(x-1)^2)=int(1dx)/x^2+int(2dx)/x-int(2dx)/(x-1)+ (7dx)/(x-1)^2#

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To integrate (6x^2 + 1)/(x^2(x - 1)^2) using partial fractions, first, express the rational function as a sum of partial fractions.

The partial fraction decomposition of (6x^2 + 1)/(x^2(x - 1)^2) is:

(6x^2 + 1)/(x^2(x - 1)^2) = A/x + B/x^2 + C/(x - 1) + D/(x - 1)^2

To find the values of A, B, C, and D, you can multiply both sides by the common denominator x^2(x - 1)^2:

6x^2 + 1 = A(x - 1)^2 + Bx(x - 1)^2 + Cx^2(x - 1) + Dx^2

After expanding and simplifying, equate coefficients of like terms to find A, B, C, and D.

Once you have found the values of A, B, C, and D, rewrite the original expression as a sum of the partial fractions. Then, integrate each term separately.

The integral of A/x is A*ln|x|, the integral of B/x^2 is -B/x, the integral of C/(x - 1) is C*ln|x - 1|, and the integral of D/(x - 1)^2 is -D/(x - 1).

Finally, combine the integrals of the partial fractions and add the constant of integration, if necessary, to obtain the final result.

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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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