# How do you integrate #int (6(5 - x))/( (x - 7)(4 - x))# using partial fractions?

When the numerators coincide,

Thus, we have

#int(6(5-x))/((x-7)(4-x))dx =int(4/(x-7)+2/(x-4) )dx#

According to Log Rule,

I hope this made sense.

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To integrate ( \int \frac{{6(5 - x)}}{{(x - 7)(4 - x)}} ) using partial fractions, you'll first decompose the fraction into simpler fractions using partial fraction decomposition.

- Decompose ( \frac{{6(5 - x)}}{{(x - 7)(4 - x)}} ) into partial fractions.
- Find the constants ( A ) and ( B ) such that ( \frac{{6(5 - x)}}{{(x - 7)(4 - x)}} = \frac{A}{{x - 7}} + \frac{B}{{4 - x}} ).
- Clear the fractions and solve for ( A ) and ( B ).
- Integrate each partial fraction term separately.

Following these steps:

[ \begin{align*} \frac{{6(5 - x)}}{{(x - 7)(4 - x)}} &= \frac{A}{{x - 7}} + \frac{B}{{4 - x}} \ 6(5 - x) &= A(4 - x) + B(x - 7) \ &= (A - B)x + (4A - 7B) \end{align*} ]

Matching coefficients: [ 5(6) = 4A - 7B ] [ -6 = A - B ]

Solving these equations gives ( A = -1 ) and ( B = -5 ).

So, ( \frac{{6(5 - x)}}{{(x - 7)(4 - x)}} = \frac{{-1}}{{x - 7}} + \frac{{-5}}{{4 - x}} ).

Now, integrate each term separately: [ \int \frac{{-1}}{{x - 7}} , dx = -\ln|x - 7| + C_1 ] [ \int \frac{{-5}}{{4 - x}} , dx = -5\ln|4 - x| + C_2 ]

Where ( C_1 ) and ( C_2 ) are constants of integration.

So, the integral of ( \frac{{6(5 - x)}}{{(x - 7)(4 - x)}} ) is: [ -\ln|x - 7| - 5\ln|4 - x| + C ] where ( C = C_1 + C_2 ).

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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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