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

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Take the constant out,


Now let's apply integral substitution:`u=sin(x)`



Now to use partial fractions, denominator of the integrand needs to be factored,

Let's split the middle term,




Now let's write it as sum of partial fractions:


Multiply the above by the LCD,




Equating the coefficients of the like terms,

`A+B=0`   -----------------------------(1)

`4A-B=1`  ----------------------------(2)

Solve the above linear equations to get the values of A and B,

Add equation 1 and 2,



Plug the value of A in equation 1,



Plug in the values of A and B in the partial fraction template,





Take the constant out,


Apply the sum rule,


Now use the common integral:`int1/xdx=ln|x|`


Substitute back `u=sin(x)`



Simplify and add a constant C to the solution,


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