Use integration by parts to determine: `int` e^x  sinx dx

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The question has been edited to conform to the eNotes' policy of not allowing multiple questions to be asked in a single post.

The integral `int e^x sin x dx` has to be determined.

`int e^x sin x dx`

= `e^x*sin x - int e^x*cos x dx`

= `e^x*sin x - (e^x*cos x + int e^x*sin x dx)`

= `e^x*sin x - e^x*cos x - int e^x*sin x dx`

=> `2*int e^x sin x dx = e^x*sin x - e^x*cos x`

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

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

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