What is the integral of (1+sin(2x))/(cos(x)+sin(x)) ?

The integral of (1+sin(2x))/(cos(x)+sin(x)) is sin(x)-cos(x)+C

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integral of (1+sin(2x))/(cos(x)+sin(x))

\int \frac{1 + sin ( 2 x )}{cos ( x ) + sin ( x )} d x

sin (x) - cos (x) + C

Solution steps

\int \frac{1 + sin ( 2 x )}{cos ( x ) + sin ( x )} d x

Rewrite using trig identities

= \int cos (x) + sin (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int cos (x) d x + \int sin (x) d x

\int cos (x) d x = sin (x)

\int sin (x) d x = - cos (x)

= sin (x) - cos (x)

Add a constant to the solution

= sin (x) - cos (x) + C

\int \frac{g ( x )}{x} d x

\int \frac{sec ( x )}{2 sec ^{2} ( x ) - 1} d x

\int (x^{2} - 1)^{- \frac{3}{2}} d x

\int \frac{1}{2} + x^{4} + \frac{1}{16 x ^{4}} d x

\int \frac{1}{( a ^{2} + r ^{2} ) ^{3}} d r

What is the integral of (1+sin(2x))/(cos(x)+sin(x)) ?
The integral of (1+sin(2x))/(cos(x)+sin(x)) is sin(x)-cos(x)+C