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

The integral of e^x*cos(2)e^x is 1/2 cos(2)e^{2x}+C

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integral of e^x*cos(2)e^x

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

\frac{1}{2} cos (2) e^{2 x} + C

Solution steps

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

Take the constant out:

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

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

Apply exponent rule:

a^{n} \cdot b^{n} = (a \cdot b)^{n}

e^{x} e^{x} = (ee)^{x}

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

\int a^{x} d x = \frac{a ^{x}}{ln a}

= cos (2) \frac{( ee ) ^{x}}{ln ( ee )}

Simplify

cos (2) \frac{( ee ) ^{x}}{ln ( ee )} : \frac{1}{2} cos (2) e^{2 x}

= \frac{1}{2} cos (2) e^{2 x}

Add a constant to the solution

= \frac{1}{2} cos (2) e^{2 x} + C

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What is the integral of e^x*cos(2)e^x ?
The integral of e^x*cos(2)e^x is 1/2 cos(2)e^{2x}+C