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

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

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integral of 2cos(x)e^x

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

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

Solution steps

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

\int 10 e^{1 - 5 x} d x

\frac{\partial}{\partial y} (3 x sin (y))

\int 7 x^{\frac{2}{5}} + 2 x^{- \frac{4}{5}} d x

\frac{d}{d x} (tan (x^{2} + y))

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

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