`int 5/(x^2+3x-4) dx` Use partial fractions to find the indefinite integral

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`int 5/(x^2+3x-4)dx`

To solve using partial fraction, the denominator of the integrand should be factored.


Then, express it as sum of fractions.


To determine the values of A and B, multiply both sides by the LCD of the fractions present.



Then, assign values to x in which either x+4 or x-1 will become zero.

So plug-in x=-4 to get the value of A.





Also, plug-in x=1





So the partial fraction decomposition of the integrand is

`int 5/(x^2+3x-4)dx`

`= int 5/((x+4)(x-1))dx`

`= int (-1/(x+4)+1/(x-1))dx`

Then, express it as two integrals.

`= int -1/(x+4)dx + int 1/(x-1)dx`

`= - int 1/(x+4)+int 1/(x-1)dx`

To take the integral, apply the formula `int 1/u du = ln|u| + C` .

`= -ln|x+4| + ln|x-1| + C`


Therefore, `int 5/(x^2+3x-4)dx= -ln|x+4| + ln|x-1| + C` .

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