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

Feb 14, 2018

$\frac{5}{2} x - \frac{1}{2} \ln \left(\cos x + \sin x\right) + C$

#### Explanation:

Write the numerator in the form

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

This means that

$a + b = 2 , \quad a - b = 3 \implies a = \frac{5}{2} , b = - \frac{1}{2}$

Thus

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

This can be easily integrated to
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$\frac{5}{2} x - \frac{1}{2} \ln \left(\cos x + \sin x\right) + C$