integrate by parts ...

Question

darkigloo · Answer

$$\int\limits e ^{6x}\sin(5x)dx$$

darkigloo · Answer

is this correct? 
$$u =\sin(5x) , du = 5 \cos(5x) , dv = e ^{6x}dx , v = \frac{ e ^{6x} }{ 6}$$

kainui · Answer

Yeah that's right, keep going!

This is a tricky one, but I think it's pretty fun once you finish it. You have to do integration by parts twice, because sine ends up becoming a multiple of its own derivative after you take the derivative of sine twice.

darkigloo · Answer

$$=\frac{ e ^{6x}\sin(5x) }{ 6 } - \frac{ 5 }{ 6 } \int\limits \cos(5x)e ^{6x} dx$$

darkigloo · Answer

and then i did u= e^6x , du = 6e^6x , dv= cos(5x), v = sin(5x)

darkigloo · Answer

$$= \frac{ e ^{6x}\sin(5x) }{ 6 } - \left[ \frac{ 5 }{ 6 } e ^{6x}\sin(5x)-\int\limits \sin(5x)6e ^{6x}dx\right]$$

kainui · Answer

This is in the right kind of direction, but you need to be consistent in your choices, so you should pick the more similar: (otherwise you are going backwards the same step instead of forwards two steps)

$$u =\cos(5x) , du = -5 \sin(5x) , dv = e ^{6x}dx , v = \frac{ e ^{6x} }{ 6}$$