What is the integral of 2(cos(e^x)-e^xxsin(e^x)) ?

The integral of 2(cos(e^x)-e^xxsin(e^x)) is 2xcos(e^x)+C

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integral of 2(cos(e^x)-e^xxsin(e^x))

\int 2 (cos (e^{x}) - e^{x} x sin (e^{x})) d x

2 x cos (e^{x}) + C

Solution steps

\int 2 (cos (e^{x}) - e^{x} x sin (e^{x})) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int cos (e^{x}) - e^{x} x sin (e^{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

= 2 (\int cos (e^{x}) d x - \int e^{x} x sin (e^{x}) d x)

\int cos (e^{x}) d x = Ci (e^{x})

\int e^{x} x sin (e^{x}) d x = - x cos (e^{x}) + Ci (e^{x})

= 2 (Ci (e^{x}) - (- x cos (e^{x}) + Ci (e^{x})))

Simplify

= 2 x cos (e^{x})

Add a constant to the solution

= 2 x cos (e^{x}) + C

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What is the integral of 2(cos(e^x)-e^xxsin(e^x)) ?
The integral of 2(cos(e^x)-e^xxsin(e^x)) is 2xcos(e^x)+C