What is the integral of (esin^2(x)+3e^x+ecos^2(x)) ?

The integral of (esin^2(x)+3e^x+ecos^2(x)) is ex+3e^x+C

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integral of (esin^2(x)+3e^x+ecos^2(x))

\int (e sin^{2} (x) + 3 e^{x} + e cos^{2} (x)) d x

e x + 3 e^{x} + C

Solution steps

\int e sin^{2} (x) + 3 e^{x} + e cos^{2} (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 sin^{2} (x) d x + \int 3 e^{x} d x + \int e cos^{2} (x) d x

\int e sin^{2} (x) d x = \frac{e}{2} (x - \frac{1}{2} sin (2 x))

\int 3 e^{x} d x = 3 e^{x}

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

= \frac{e}{2} (x - \frac{1}{2} sin (2 x)) + 3 e^{x} + \frac{e}{2} (x + \frac{1}{2} sin (2 x))

Simplify

\frac{e}{2} (x - \frac{1}{2} sin (2 x)) + 3 e^{x} + \frac{e}{2} (x + \frac{1}{2} sin (2 x)) : e x + 3 e^{x}

= e x + 3 e^{x}

Add a constant to the solution

= e x + 3 e^{x} + C

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What is the integral of (esin^2(x)+3e^x+ecos^2(x)) ?
The integral of (esin^2(x)+3e^x+ecos^2(x)) is ex+3e^x+C