What is the integral of (asin(nx)+bcos(mx)) ?

The integral of (asin(nx)+bcos(mx)) is b 1/m sin(mx)-a 1/n cos(nx)+C

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integral of (asin(nx)+bcos(mx))

\int (a sin (n x) + b cos (m x)) d x

b \frac{1}{m} sin (m x) - a \frac{1}{n} cos (n x) + C

Solution steps

\int a sin (n x) + b cos (m 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 (m x) b d x + \int sin (n x) a d x

\int cos (m x) b d x = b \frac{1}{m} sin (m x)

\int sin (n x) a d x = - a \frac{1}{n} cos (n x)

= b \frac{1}{m} sin (m x) - a \frac{1}{n} cos (n x)

Add a constant to the solution

= b \frac{1}{m} sin (m x) - a \frac{1}{n} cos (n x) + C

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What is the integral of (asin(nx)+bcos(mx)) ?
The integral of (asin(nx)+bcos(mx)) is b 1/m sin(mx)-a 1/n cos(nx)+C