Monday, August 5, 2013

`int t sec^2 2t dt` Evaluate the integral

`inttsec^2(2t)dt` 


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


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


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


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


Now using the above method of integration by parts,


`inttsec^2(2t)dt=tintsec^2(2t)dt-int(d/dt(t)intsect^2(2t))dt`


`=t*tan(2t)/2-int(1*tan(2t)/2)dt`


`=1/2t*tan(2t)-1/2inttan(2t)dt`


Now let's evaluate `inttan(2t)dt` by using the method of substitution,


Substitute `x=cos(2t),=>dx=-2sin(2t)dt`


`inttan(2t)dt=int(sin(2t)/cos(2t))dt`


`=intdx/(-2x)`


`=-1/2ln|x|`


substitute back `x=cos(2t)`


`=-1/2ln|cos(2t)|`


`inttsec^2(2t)dt=1/2t*tan(2t)-1/2(-1/2ln|cos(2t)|+C`


C is a constant


`inttsec^2(2t)dt=1/2t*tan(2t)+1/4ln|cos(2t)|+C`


`inttsec^2(2t)dt` 


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


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


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


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


Now using the above method of integration by parts,


`inttsec^2(2t)dt=tintsec^2(2t)dt-int(d/dt(t)intsect^2(2t))dt`


`=t*tan(2t)/2-int(1*tan(2t)/2)dt`


`=1/2t*tan(2t)-1/2inttan(2t)dt`


Now let's evaluate `inttan(2t)dt` by using the method of substitution,


Substitute `x=cos(2t),=>dx=-2sin(2t)dt`


`inttan(2t)dt=int(sin(2t)/cos(2t))dt`


`=intdx/(-2x)`


`=-1/2ln|x|`


substitute back `x=cos(2t)`


`=-1/2ln|cos(2t)|`


`inttsec^2(2t)dt=1/2t*tan(2t)-1/2(-1/2ln|cos(2t)|+C`


C is a constant


`inttsec^2(2t)dt=1/2t*tan(2t)+1/4ln|cos(2t)|+C`


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