∫(sin 5x)dx = ∫(sin 5x)/5 d(5x) = -cos(5x)/5 + c
2007-01-04 18:53:52
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answer #1
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answered by Anonymous
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Integral (sin5x) dx
Rule of thumb: Every time you're taking the integral of a function with its insides being linear (in this case, 5x is linear since x is a power of 1), it is easy to do mental substitution.
All you have to do is take the integral of sin(x) mentally (the integral of sinx is -cosx), and apply it with the 5x (so we would get -cos5x). However, when you take the derivative of -cos(5x), you would get 5sin(5x), because the chain rule would force you to multiply 5. To offset this, you merely multiply by (1/5) to the integral. Therefore
Integral (sin(5x))dx = (-1/5)cos(5x) + C
If you'd like to actually see the substitution being done, I'll show you with u substitution.
Integral (sin(5x))dx
Let u = 5x
du = 5 dx
(1/5) du = dx
Therefore,
Integral (sin(5x))dx = Integral (sin(u) (1/5)du)
Pulling out the constants out of the integral, we get
(1/5) * Integral (sin(u)du)
Integrating appropriately, we have
(1/5) [-cos(u)] + C
(-1/5) cos(u) + C
Replacing u = 5x,
(-1/5) cos(5x) + C
2006-12-31 19:50:22
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answer #2
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answered by Puggy 7
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â«(sin 5x)dx = â«(sin 5x)/5 d(5x) = -cos(5x)/5 + c
Here, I used mental substitution. If you don't like it, you can substitute u = 5x. Either way, you should get the same answer.
2006-12-31 19:23:53
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answer #3
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answered by sahsjing 7
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(-(cos 5x)/5)+c
2006-12-31 19:45:33
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answer #4
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answered by Alex M 2
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-(cos5x)/5 +C. if it was 'a' instead of 5, then it would have been -(cosax)/a +C
2007-01-01 00:17:39
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answer #5
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answered by IN PURSUIT OF WISDOM 2
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it can be solved by the reduction formula
2006-12-31 19:25:02
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answer #6
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answered by john d 1
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