Evaluate the integral: `int e^(2x)*sin(3x) dx`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The integral `int e^(2x)*sin(3x) dx` has to be evaluated.

Use integration by parts: `int u dv = u*v - int v*du`

`e^(2*x) = u` , `dv = sin (3*x) dx`

`du = 2*e^(2*x) dx` , `v = -cos(3*x)/3`

`int e^(2x)*sin(3x) dx`

=> `-e^(2x)*cos(3x)/3 + (2/3)*int cos(3*x) e^(2*x) dx`

Again use integration by parts for `int cos(3*x) e^(2*x) dx`

`u = e^(2x)` , `dv = cos (3x) dx`

`du = 2*e^(2x) dx` , `v = sin(3x)/3`

=> `e^(2x)*sin(3x)/3 - int sin(3x)/3 2*e^(2x) dx`

=> `e^(2x)*sin(3x)/3 - (2/3)*int sin(3x)e^(2x) dx`

`int e^(2x)*sin(3x) dx` = `-e^(2x)*cos(3x)/3 + (2/3)(e^(2x)*sin(3x)/3) - (4/9)*int sin(3x)e^(2x) dx`

=> `(13/9)int sin(3x)e^(2x) dx` = `-e^(2x)*cos(3x)/3 + (2/3)(e^(2x)*sin(3x)/3)`

=> `int sin(3x)e^(2x) dx` = `(-3*e^(2x)cos(3x) + 2*e^(2x)*sin(3x))/13`

The integral `int e^(2x)*sin(3x) dx` = `(2*e^(2x)*sin (3x)- 3*e^(2x)cos(3x))/13`

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