What is the integral of 2e^xcos(x) ?

The integral of 2e^xcos(x) is e^xsin(x)+e^xcos(x)+C

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integral of 2e^xcos(x)

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Simplify

2 (\frac{e ^{x} sin ( x )}{2} + \frac{e ^{x} cos ( x )}{2}) : e^{x} sin (x) + e^{x} cos (x)

= e^{x} sin (x) + e^{x} cos (x)

Add a constant to the solution

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

\frac{\partial}{\partial y} (e^{y})

\frac{\partial}{\partial x} (4 x y - x^{4} - y^{4} + 1)

\int_{0}^{x} e^{- t^{2}} d t

\frac{x + 1}{4 x ^{2} + 12 x + 2}

\frac{d}{d x} (3 arcsin (x^{2}))

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