What is the integral of cos(x)(sin(x)+7)e^{sin(x)} ?

The integral of cos(x)(sin(x)+7)e^{sin(x)} is e^{sin(x)}sin(x)+6e^{sin(x)}+C

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integral of cos(x)(sin(x)+7)e^{sin(x)}

\int cos (x) (sin (x) + 7) e^{s i n (x)} d x

e^{s i n (x)} sin (x) + 6 e^{s i n (x)} + C

Solution steps

\int cos (x) (sin (x) + 7) e^{s i n (x)} d x

Expand

cos (x) (sin (x) + 7) e^{s i n (x)} : e^{s i n (x)} cos (x) sin (x) + 7 e^{s i n (x)} cos (x)

= \int e^{s i n (x)} cos (x) sin (x) + 7 e^{s i n (x)} cos (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 e^{s i n (x)} cos (x) sin (x) d x + \int 7 e^{s i n (x)} cos (x) d x

\int e^{s i n (x)} cos (x) sin (x) d x = sin (x) e^{s i n (x)} - e^{s i n (x)}

\int 7 e^{s i n (x)} cos (x) d x = 7 e^{s i n (x)}

= sin (x) e^{s i n (x)} - e^{s i n (x)} + 7 e^{s i n (x)}

Simplify

= e^{s i n (x)} sin (x) + 6 e^{s i n (x)}

Add a constant to the solution

= e^{s i n (x)} sin (x) + 6 e^{s i n (x)} + C

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What is the integral of cos(x)(sin(x)+7)e^{sin(x)} ?
The integral of cos(x)(sin(x)+7)e^{sin(x)} is e^{sin(x)}sin(x)+6e^{sin(x)}+C