How do you integrate #(2x1)/((x+1)(x2)(x+3))# using partial fractions?
Before integration, partial fractions can be done as explained below
The partial fractions would thus be
Integration is now simple
=
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To integrate ( \frac{2x  1}{(x + 1)(x  2)(x + 3)} ) using partial fractions, follow these steps:

First, express the rational function as the sum of partial fractions: [ \frac{2x  1}{(x + 1)(x  2)(x + 3)} = \frac{A}{x + 1} + \frac{B}{x  2} + \frac{C}{x + 3} ]

Multiply both sides by the denominator ( (x + 1)(x  2)(x + 3) ) to clear the fractions: [ 2x  1 = A(x  2)(x + 3) + B(x + 1)(x + 3) + C(x + 1)(x  2) ]

Expand and equate coefficients of like terms. This will give you a system of linear equations to solve for ( A ), ( B ), and ( C ).

Once you find the values of ( A ), ( B ), and ( C ), rewrite the original function using these partial fractions.

Integrate each partial fraction separately.

Finally, add the integrated partial fractions together to obtain the final result.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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