How do you integrate #int 1/((3 + x) (1  x)) # using partial fractions?
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To integrate ( \frac{1}{{(3 + x)(1  x)}} ) using partial fractions, follow these steps:

Express the given fraction as a sum of simpler fractions: ( \frac{1}{{(3 + x)(1  x)}} = \frac{A}{3 + x} + \frac{B}{1  x} )

Find the values of ( A ) and ( B ) by equating numerators: ( 1 = A(1  x) + B(3 + x) )

Solve for ( A ) and ( B ) by choosing suitable values of ( x ). Typically, choose ( x = 1 ) and ( x = 3 ) to simplify the equation.

Once you find the values of ( A ) and ( B ), rewrite the original fraction with the partial fraction decomposition.

Integrate each term separately using basic integration techniques.

Finally, combine the integrals to obtain the overall integral of the original expression.

The integral of ( \frac{1}{{(3 + x)(1  x)}} ) using partial fractions will yield the solution.
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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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