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In this problem we are going to compute the following integral.
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Integral dx 6x squared plus 8x over x cubed plus x squared plus x plus 1.
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Now let us attack the expression in the denominator.
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Because if we manage to simplify that expression, we will be able to use the partial fraction decomposition, which will make our lives easier.
00:41
So we have 6x squared plus 8x divided by.
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Okay, let us consider the first two terms.
00:52
We can factor out x squared to write the first two terms.
01:02
Plus we have the rest of the terms x plus 1 and now we see the following integral d x 6 x squared plus 8 x over x squared plus 1 times x plus 1 now i'm going to make a remark here and i'll follow from this line shortly after performing this partial fraction decomposition.
01:38
Okay, so let us consider this integrant 6x squared plus 8x over x squared plus 1 times x plus 1.
01:54
So we have two factors in the denominator, x squared plus 1 and x plus 1.
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So we should be able to separate this, separate the expression on the left -hand side into two expressions.
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One of them will contain x squared plus one in the denominator and the other one will contain x plus one.
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And the numerators, of course, should have, should be polynomials of order one less than those in the denominator.
02:33
So we expect to have some linear function here in the numerator of the first term and just some constant function in the numerator of the second term.
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Now the rest is just the determination of these unknown coefficients, a, b and c.
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So we start with equating the denominator.
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So i'm going to multiply the first term by x plus.
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And the second turn by x squared plus one.
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So i get a x plus b times x plus one plus c times x squared plus one everything divided by x squared plus one times x plus one...