What is the solution to the integral of 2cosx + 3sinx / cosx + sinx? Thanks

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Answer 1 · 2018-02-14T11:02:31

$5/2 x -1/2 ln (cos x + sin x) + C$

Write the numerator in the form

$2 cos x+ 3sin x = a(cos x+ sin x) + b (cos x - sin x)$

This means that

$a+b=2, quad a-b =3 implies a=5/2, b=-1/2$

Thus

${2 cos x + 3 sin x }/{ cos x + sin x }= 5/2 -1/2 {cos x - sin x }/ {cos x + sin x}$

This can be easily integrated to
h
$5/2 x -1/2 ln (cos x + sin x) + C$