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Evaluate the integral.

$ \displaystyle \int_2^4 \frac{x + 2}{x^2 + 3x - 4}\ dx $

$$\frac{1}{5} \ln 48$$

Integration Techniques

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for this problem. Let's try a partial freshen decomposition. So if we will use that, the first thing to check and the denominator is whether or not this full factor. So if it does, we must factor it first before using integration before doing partial fraction the composition so we can rewrite the integration is this and using what the author calls case one for a partial freshens. This is when you have distinct linear factors in the denominator and they're not repeated. So both her to the first power. Could you put a one there? Then we have a over X plus four be over X minus one God and multiply both sides of this equation by the denominator on the left, we have a X minus one, and then we have the extras for and we could go ahead and pull out of X. And then there's our constant term. So by comparing the coefficients on the left than on the right, we could see that a plus B must equal one. So that's one equation, and from looking at the constants are we have four B minus A, and that must equal to silly me rewrite that So here this is just the two by two system that we can solve for Andy. For example, if I just add these two equations together on the left, five B on the right three that gives me B and then plugging this back into this equation up here, we find that a equals two over five. So now we have R and R B with plug these in. And then, instead of integrating the original function, will just integrate this partial fraction here with Andy. So I'm running out of room here. Let me go to the next base for that. So plugging in and be are integral becomes we have to over five for a and then we have three over five for B. Now, you may have memorized how to integrate this by now, but if this plus four or if this minus one are bothering you, you could split this into two in a girls and then you would have to do to separate you subs. So that would be for the first in a role. And then here you have another integral And this time you would do u equals X minus one. No. So In any case, two will revive natural law. Deathless for plus three over five. Natural log X minus one and your end points two, two four and delicious plug goes in. So plug in the four. First two over five. Natural log eight plus three over fires Natural Lago three. Thanks. Now we plug it into two to over five. Ellen six and then three over five. Ellen of one and that zero. So we can ignore that, Sir Melon of one. Now we just combine, combined these three terms here and simple by. And this all reduces too. And when a forty eight all divided by five help and that's your final answer.