What is the integral of e^{2x}4cos(2x) ?

The integral of e^{2x}4cos(2x) is e^{2x}sin(2x)+e^{2x}cos(2x)+C

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integral of e^{2x}4cos(2x)

\int e^{2 x} 4 cos (2 x) d x

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

Solution steps

\int e^{2 x} \cdot 4 cos (2 x) d x

Take the constant out:

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

= 4 \cdot \int e^{2 x} cos (2 x) d x

Apply Integration By Parts

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

= 4 (\frac{1}{2} e^{2 x} sin (2 x) - \int e^{2 x} sin (2 x) d x)

Apply Integration By Parts

: - \frac{1}{2} e^{2 x} cos (2 x) - \int - e^{2 x} cos (2 x) d x

= 4 (\frac{1}{2} e^{2 x} sin (2 x) - (- \frac{1}{2} e^{2 x} cos (2 x) - \int - e^{2 x} cos (2 x) d x))

Take the constant out:

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

= 4 (\frac{1}{2} e^{2 x} sin (2 x) - (- \frac{1}{2} e^{2 x} cos (2 x) - (- \int e^{2 x} cos (2 x) d x)))

Therefore

4 \cdot \int e^{2 x} cos (2 x) d x = 4 (\frac{1}{2} e^{2 x} sin (2 x) - (- \frac{1}{2} e^{2 x} cos (2 x) - (- \int e^{2 x} cos (2 x) d x)))

Isolate

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

= 4 (\frac{e ^{2 x} sin ( 2 x )}{4} + \frac{e ^{2 x} cos ( 2 x )}{4})

Simplify

4 (\frac{e ^{2 x} sin ( 2 x )}{4} + \frac{e ^{2 x} cos ( 2 x )}{4}) : e^{2 x} sin (2 x) + e^{2 x} cos (2 x)

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

Add a constant to the solution

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

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What is the integral of e^{2x}4cos(2x) ?
The integral of e^{2x}4cos(2x) is e^{2x}sin(2x)+e^{2x}cos(2x)+C