# How do you find the antiderivative of #int 1/(4-x^2)dx#?

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To find the antiderivative of ( \int \frac{1}{4-x^2} , dx ), you can use the technique of partial fraction decomposition. First, factor the denominator as ( (2+x)(2-x) ). Then, express ( \frac{1}{4-x^2} ) as the sum of two fractions with undetermined constants:

[ \frac{1}{4-x^2} = \frac{A}{2+x} + \frac{B}{2-x} ]

Solve for ( A ) and ( B ) by equating numerators:

[ 1 = A(2-x) + B(2+x) ]

[ 1 = (2A + 2B) + (-A + B)x ]

Equating coefficients gives:

[ 2A + 2B = 1 ] [ -A + B = 0 ]

Solving this system of equations yields ( A = \frac{1}{4} ) and ( B = \frac{1}{4} ).

Thus, we can rewrite the integral as:

[ \int \frac{1}{4-x^2} , dx = \int \left( \frac{\frac{1}{4}}{2+x} + \frac{\frac{1}{4}}{2-x} \right) , dx ]

Now integrate each term separately:

[ \int \frac{1}{4-x^2} , dx = \frac{1}{4} \ln|2+x| - \frac{1}{4} \ln|2-x| + C ]

Where ( C ) is the constant of integration.

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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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