`int e^(4x)cos(2x) dx` Find the indefinite integral

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`int e^(4x)cos(2x)dx`

To solve, apply integration by parts `int u dv = u*v - int vdu` .

In the given integral, the let the u and dv be:

`u = e^(4x) `  

`dv = cos(2x)dx`

Then, take the derivative of u to get du. Also, take the integral of...

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`int e^(4x)cos(2x)dx`

To solve, apply integration by parts `int u dv = u*v - int vdu` .

In the given integral, the let the u and dv be:

`u = e^(4x) `  

`dv = cos(2x)dx`

Then, take the derivative of u to get du. Also, take the integral of dv to get v.

`du = e^(4x)*4dx`

`du = 4e^(4x)dx`

`intdv = int cos(2x)dx`

`v = (sin(2x))/2`

Substituting them to the integration by parts formula yields

`int e^(4x)cos(2x)dx= e^(4x)*(sin(2x))/2 - int (sin(2x))/2 * 4e^(4x)dx`

`int e^(4x)cos(2x)dx= (e^(4x)sin(2x))/2 - int 2e^(4x)sin(2x)dx`

For the integral at the right side, apply integration by parts again. Let the u and dv be:

`u = 2e^(4x)`

`dv = sin(2x)dx`

Take the derivative of u and take the integral of dv to get du and v, respectively.

`du = 2e^(4x)*4dx`

`du = 8e^(4x)dx`

`int dv = int sin(2x)dx`

`v = -cos(2x)/2`

Plug-in them to the formula of integration by parts.

`int e^(4x)cos(2x)dx= (e^(4x)sin(2x))/2 - int 2e^(4x)sin(2x)dx`

`int e^(4x)cos(2x)dx= (e^(4x)sin(2x))/2 - [2e^(4x)*(-(cos(2x))/2) - int (-(cos(2x))/2)*8e^(4x)dx]`

`int e^(4x)cos(2x)dx= (e^(4x)sin(2x))/2 - [-e^(4x)cos(2x)+int 4e^(4x)cos(2x)dx]`

`int e^(4x)cos(2x)dx= (e^(4x)sin(2x))/2 +e^(4x)cos(2x) - 4int e^(4x)cos(2x)dx`

Since the integrals at the left and right side of the equation are like terms, bring them together on one side.

`int e^(4x)cos(2x)dx+4int e^(4x)cos(2x)dx= (e^(4x)sin(2x))/2 +e^(4x)cos(2x)`

The left side simplifies to

`5int e^(4x)cos(2x)dx= (e^(4x)sin(2x))/2 +e^(4x)cos(2x)`

Isolating the integral, the equation becomes

`int e^(4x)cos(2x)dx= ((e^(4x)sin(2x))/2 +e^(4x)cos(2x)) * 1/5`

`int e^(4x)cos(2x)dx= (e^(4x)sin(2x))/10 +(e^(4x)cos(2x))/5`

Since it is an indefinite integral, add C at the right side.

Therefore, 

`int e^(4x)cos(2x)dx= (e^(4x)sin(2x))/10 +(e^(4x)cos(2x))/5+C ` .

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