Sunday, July 26, 2015

`int e^(2 theta) sin(3 theta) d theta` Evaluate the integral

`inte^2thetasin(3theta)d theta`


If f(x) and g(x) are differentiable functions, then


`intf(x)g'(x)=f(x)g(x)-intf'(x)g(x)dx`


If we rewrite f(x)=u and g'(x)=v, then


`intuvdx=uintvdx-int(u'intvdx)dx`


Let's integrate using the above method of integration by parts,


Let `u=e^2theta, v=sin(3theta)`


`inte^(2theta)sin(3theta)d theta=e^(2theta)intsin(3theta)d theta-int(d/(d theta)(e^(2theta))intsin(3theta)d theta)d theta`


`=e^(2theta)(-1/3cos(3theta)-int(e^(2theta)2(-1/3cos(3theta))d theta`


`=-1/3e^(2theta)cos(3theta)+2/3inte^(2theta)cos(3theta)d theta` 


apply again integration by parts,


`=-1/3e^(2theta)cos(3theta)+2/3(e^(2theta)*intcos(3theta)d theta-int(e^(2theta)*2intcos(3theta) d theta)`


`=-1/3e^(2theta)cos(3theta)+2/3(e^(2theta)1/3sin(3theta)-int2e^(2theta)(sin(3theta)/3)d theta)`


`=-1/3e^(2theta)cos(3theta)+2/9e^(2theta)sin(3theta)-4/9inte^(2theta)sin(3theta)d theta`


Isolate `inte^(2theta)sin(3theta)d theta`


`(1+4/9)inte^(2theta)sin(3theta)d theta=-1/3e^(2theta)cos(3theta)+2/9e^(2theta)sin(3theta)`


`inte^(2theta)sin(3theta)d theta=9/13(-1/3e^(2theta)cos(3theta)+2/9e^(2theta)sin(3theta))`


`=-3/13e^(2theta)cos(3theta)+2/13e^(2theta)sin(3theta)`


add a constant C to the solution,


`=-3/13e^(2theta)cos(3theta)+2/13e^(2theta)sin(3theta)+C`


`inte^2thetasin(3theta)d theta`


If f(x) and g(x) are differentiable functions, then


`intf(x)g'(x)=f(x)g(x)-intf'(x)g(x)dx`


If we rewrite f(x)=u and g'(x)=v, then


`intuvdx=uintvdx-int(u'intvdx)dx`


Let's integrate using the above method of integration by parts,


Let `u=e^2theta, v=sin(3theta)`


`inte^(2theta)sin(3theta)d theta=e^(2theta)intsin(3theta)d theta-int(d/(d theta)(e^(2theta))intsin(3theta)d theta)d theta`


`=e^(2theta)(-1/3cos(3theta)-int(e^(2theta)2(-1/3cos(3theta))d theta`


`=-1/3e^(2theta)cos(3theta)+2/3inte^(2theta)cos(3theta)d theta` 


apply again integration by parts,


`=-1/3e^(2theta)cos(3theta)+2/3(e^(2theta)*intcos(3theta)d theta-int(e^(2theta)*2intcos(3theta) d theta)`


`=-1/3e^(2theta)cos(3theta)+2/3(e^(2theta)1/3sin(3theta)-int2e^(2theta)(sin(3theta)/3)d theta)`


`=-1/3e^(2theta)cos(3theta)+2/9e^(2theta)sin(3theta)-4/9inte^(2theta)sin(3theta)d theta`


Isolate `inte^(2theta)sin(3theta)d theta`


`(1+4/9)inte^(2theta)sin(3theta)d theta=-1/3e^(2theta)cos(3theta)+2/9e^(2theta)sin(3theta)`


`inte^(2theta)sin(3theta)d theta=9/13(-1/3e^(2theta)cos(3theta)+2/9e^(2theta)sin(3theta))`


`=-3/13e^(2theta)cos(3theta)+2/13e^(2theta)sin(3theta)`


add a constant C to the solution,


`=-3/13e^(2theta)cos(3theta)+2/13e^(2theta)sin(3theta)+C`


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