What is the integral of sin(ax)-e^{x/b} ?

The integral of sin(ax)-e^{x/b} is -1/a cos(ax)-be^{x/b}+C

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integral of sin(ax)-e^{x/b}

\int sin (a x) - e^{\frac{x}{b}} d x

- \frac{1}{a} cos (a x) - b e^{\frac{x}{b}} + C

Solution steps

\int sin (a x) - e^{\frac{x}{b}} 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 sin (a x) d x - \int e^{\frac{x}{b}} d x

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

\int e^{\frac{x}{b}} d x = b e^{\frac{x}{b}}

= - \frac{1}{a} cos (a x) - b e^{\frac{x}{b}}

Add a constant to the solution

= - \frac{1}{a} cos (a x) - b e^{\frac{x}{b}} + C

\int x (x + 5) d x

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What is the integral of sin(ax)-e^{x/b} ?
The integral of sin(ax)-e^{x/b} is -1/a cos(ax)-be^{x/b}+C