Integrate the following using Partial Fractions. ∫xdxx2−3x−4

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Integrate the following using Partial Fractions.   ∫xdxx2−3x−4

zesponderyd · Accepted Answer

Given: ∫xdxx2−3x−4  To Find-  The value of the above integration using partial fractions.  Explanation- Rewrite the given expression as,  I=∫xdxx2−3x−4  Now, simplifying the above expression as follows,we get,  =∫xdxx4−4x+x−4  =∫x(x+1)(x−4)dx  The above expression can be simplified using partial fractions, we get,  x(x+1)(x−4)=A(x+1)+B(x−4)…(1)  x(x+1)(x−4)=A(x−4)+B(x+1)(x+1)(x−4)  x=A(x−4)+B(x+1)  To Find the value of A and B, we have to substitute the value of x as 4 in the above expression, we get,  4=0+5⋅B  Further, we can write as follows,  B=45  Similarily, substituting the value of x as -1 in the above expression, we get,  A=15  So, the equation (1) can be written as follows,  ∫x(x+1)(x−4)=15∫1(x+1)dx+45∫1(x−4)dx  =15ln|x+1|+45ln|x−4|+C  Answer- Hence, the solution of the integral ∫xdxx2−3x−4  is  15ln|x+1|+45ln|x−4|+C

kaluitagf · Accepted Answer

∫xx2−3x−4dx  Take the partial fractions of : xx2−3x−4:15(x+1)+45(x−4)  =∫15(x+1)+45(x−4)dx  Apply the Sum Rule: ∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx  =∫15(x+1)dx+45(x−4)dx  ∫15(x+1)dx=15ln|x+1|  ∫45(x−4)dx=45ln|x−4|  =15ln|x+1|+45ln|x−4|  Solution:  =15ln|x+1|+45ln|x−4|+C

RizerMix · Accepted Answer

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Integrate the following using Partial Fractions. \int \frac{xdx}{x^2-3x-4}

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