Wednesday, December 17, 2008

`int x cos(5x) dx` Evaluate the integral

`intxcos(5x)dx`


To evaluate, apply integration by parts `int udv = uv -intv du` .


So let:


`u=x `     and     `dv =intcos(5x)dx`


Differentiating u and integrating dv yield:


`du=dx `     and     `v=sin (5x)/5`


Plugging them to the formula, the integral becomes:


`int xcos(5x)dx`


`= x*sin(5x)/5 - int sin(5x)/5 dx`


`= (xsin(5x))/5 - 1/5int sin(5x)dx`


`= (xsin(5x))/5 - 1/5(-cos(5x)/5) + C`


`=(xsin(5x))/5 + cos(5x)/25+C`



Therefore, `intxcos(5x)dx=(xsin(5x))/5 + cos(5x)/25+C` .

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