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

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

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

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

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

Solution steps

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

Therefore

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

Isolate

\int e^{2 x} sin (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

\frac{( x + 1 )}{( x + 3 )}

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\int e^{10 x} (10) d x

\int x csc (tan (x^{2})) sec^{2} (x^{2}) d x

\frac{\partial}{\partial x} (x^{4} + 5 x y^{3})

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