How do you integrate #f(x)=x/((x2)(x+4)(x7))# using partial fractions?
Solving it back we have
That also means,
Or
Or, still
From there, it's just a question of solving the system. Using Cramer's rule for example; solve whichever you way you think is best. For brevity's sake I won't add more to it here, if you think you could benefit from an explanation on that, just contact me.
From there we can just say
Which are easy integrals
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To integrate ( f(x) = \frac{x}{(x2)(x+4)(x7)} ) using partial fractions, follow these steps:

Factor the denominator: ( (x2)(x+4)(x7) ) factors into ( (x2)(x+4)(x7) ).

Write ( f(x) ) as a sum of partial fractions: ( \frac{x}{(x2)(x+4)(x7)} = \frac{A}{x2} + \frac{B}{x+4} + \frac{C}{x7} )

Clear the denominators by multiplying both sides by ( (x2)(x+4)(x7) ): ( x = A(x+4)(x7) + B(x2)(x7) + C(x2)(x+4) )

Solve for ( A ), ( B ), and ( C ) by substituting values for ( x ) that eliminate two of the terms:
 Let ( x = 2 ): ( 2 = 6A ) (solving for ( A ))
 Let ( x = 4 ): ( 4 = 6B ) (solving for ( B ))
 Let ( x = 7 ): ( 7 = 9C ) (solving for ( C ))

Solve for ( A ), ( B ), and ( C ): ( A = \frac{1}{3} ), ( B = \frac{2}{3} ), ( C = \frac{7}{9} )

Rewrite the original integral with the partial fraction decomposition: ( \int \frac{x}{(x2)(x+4)(x7)} , dx = \int \left(\frac{1}{3(x2)} + \frac{2}{3(x+4)} + \frac{7}{9(x7)}\right) , dx )

Integrate each term: ( \frac{1}{3} \lnx2 + \frac{2}{3} \lnx+4  \frac{7}{9} \lnx7 + C )

Combine the terms and add the constant of integration ( C ): ( \frac{1}{3} \lnx2 + \frac{2}{3} \lnx+4  \frac{7}{9} \lnx7 + C )
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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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