# How do you integrate #int [ e^x / ((e^x -4) (e^x + 6)) ] dx# using partial fractions?

See the answer below:

Continuation:

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To integrate ( \frac{e^x}{(e^x - 4)(e^x + 6)} ) using partial fractions, you can first express the integrand as a sum of simpler fractions:

[ \frac{e^x}{(e^x - 4)(e^x + 6)} = \frac{A}{e^x - 4} + \frac{B}{e^x + 6} ]

Multiply both sides by ( (e^x - 4)(e^x + 6) ) to clear the denominators:

[ e^x = A(e^x + 6) + B(e^x - 4) ]

Expand and group like terms:

[ e^x = (A + B)e^x + 6A - 4B ]

Compare coefficients of ( e^x ) on both sides:

[ 1 = A + B ] [ 0 = 6A - 4B ]

Solve the system of equations for ( A ) and ( B ):

[ A = \frac{2}{5} ] [ B = \frac{3}{5} ]

Now that you have the values of ( A ) and ( B ), rewrite the integrand using partial fractions:

[ \frac{e^x}{(e^x - 4)(e^x + 6)} = \frac{\frac{2}{5}}{e^x - 4} + \frac{\frac{3}{5}}{e^x + 6} ]

Integrate each term separately:

[ \int \frac{e^x}{(e^x - 4)(e^x + 6)} , dx = \frac{2}{5} \int \frac{1}{e^x - 4} , dx + \frac{3}{5} \int \frac{1}{e^x + 6} , dx ]

[ = \frac{2}{5} \ln|e^x - 4| + \frac{3}{5} \ln|e^x + 6| + 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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