anonymous
  • anonymous
x cos(4x)dx u = x dv = cos 4xdx @LeibyStrauss
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
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welshfella
  • welshfella
use integration by parts u = dx ---> du = 1 dv = -cos4xdx --- > v = - (1/4) sin 4 x INT udv = uv - INT vdu
anonymous
  • anonymous
\[\int\limits_{}^{}\cos(4x)dx\] u = 4x du = 4dx 1/4du = dx \[\frac{ 1 }{ 4 }\int\limits_{}^{} \cos(u)dx = \frac{ 1 }{ 4 }\sin(4x)\] \[v = \frac{ 1 }{ 4 }\sin(4x)\]
anonymous
  • anonymous
\[\int\limits_{}^{} u dv = uv- \int\limits_{}^{} v du\]

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anonymous
  • anonymous
\[x \frac{ 1 }{ 4 } \sin(4x) - \int\limits \frac{ 1 }{ 4 } \sin(4x)\] now let's solve \[\int\limits \frac{ 1 }{ 4 } \sin(4x)\] u = 4x du = 4dx 1/4 du = dx \[\frac{ 1 }{ 4 }\int\limits \sin(u)\]
anonymous
  • anonymous
\[\frac{ 1 }{ 4 }[-\cos(u)] = -\frac{ 1 }{ 4 }\cos(4x)\]
DanJS
  • DanJS
reverse of the product rule on the derivatives [f*g] ' = f ' *g +g ' *f parts is the integration of that
anonymous
  • anonymous
@vkarwee I made a mistake somewhere.
anonymous
  • anonymous
i think you forgot to multiply
anonymous
  • anonymous
I forgot to pull out the 1/4 and then multiply by 1/4 again. I'll redo it
anonymous
  • anonymous
\[1/4\int\limits_{}^{}\sin(4x)\]
DanJS
  • DanJS
sin(4x) / 16 + x* cos(4x)/4
anonymous
  • anonymous
= \[= -\frac{ 1 }{ 16 }\sin(4x)\]
DanJS
  • DanJS
sometimes, if you can have the answer, with a calculator or whatever first, it may help you figure out the method, from the answer form
anonymous
  • anonymous
Final answer \[\frac{ x }{ 4 } \sin(4x) + \frac{ 1 }{ 16 }\cos(4x)\]
DanJS
  • DanJS
the sin and cos is mixed around,
anonymous
  • anonymous
@DanJS v = 1/4 sin(4x) \[\int\limits v du = \frac{ 1 }{ 16 } \cos(4x)\]
DanJS
  • DanJS
forgot a the minus
DanJS
  • DanJS
forget i was evfen at this question, i did the original prob x*cos(4x) to get that answer, intresting how they are related though
DanJS
  • DanJS
**x*sin(4x) is what i started with my mistake
DanJS
  • DanJS
if you need to know integral of x*sin(4x) , you know how it is related to the cos version

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