Integral of sin(x/2)cos(x/2) dx
The solution
Detail solution
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There are multiple ways to do this integral.
Method #1
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Let u=sin(2x).
Then let du=2cos(2x)dx and substitute 2du:
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The integral of a constant times a function is the constant times the integral of the function:
∫udu=2∫udu
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The integral of un is n+1un+1 when n=−1:
∫udu=2u2
So, the result is: u2
Now substitute u back in:
sin2(2x)
Method #2
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Let u=2x.
Then let du=2dx and substitute 2du:
∫2sin(u)cos(u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫sin(u)cos(u)du=2∫sin(u)cos(u)du
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Let u=cos(u).
Then let du=−sin(u)du and substitute −du:
∫(−u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫udu=−∫udu
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The integral of un is n+1un+1 when n=−1:
∫udu=2u2
So, the result is: −2u2
Now substitute u back in:
−2cos2(u)
So, the result is: −cos2(u)
Now substitute u back in:
−cos2(2x)
Method #3
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Let u=cos(2x).
Then let du=−2sin(2x)dx and substitute −2du:
∫(−2u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫udu=−2∫udu
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The integral of un is n+1un+1 when n=−1:
∫udu=2u2
So, the result is: −u2
Now substitute u back in:
−cos2(2x)
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Now simplify:
sin2(2x)
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Add the constant of integration:
sin2(2x)+constant
The answer is:
sin2(2x)+constant
The answer (Indefinite)
[src]
/
|
| /x\ /x\ 2/x\
| sin|-|*cos|-| dx = C + sin |-|
| \2/ \2/ \2/
|
/
∫sin(2x)cos(2x)dx=C+sin2(2x)
The graph
sin2(21)
=
sin2(21)
Use the examples entering the upper and lower limits of integration.