Integral of sin(2*x)/cos(2*x) 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=2x.
Then let du=2dx and substitute 2du:
∫2cos(u)sin(u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫cos(u)sin(u)du=2∫cos(u)sin(u)du
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Let u=cos(u).
Then let du=−sin(u)du and substitute −du:
∫(−u1)du
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The integral of a constant times a function is the constant times the integral of the function:
∫u1du=−∫u1du
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The integral of u1 is log(u).
So, the result is: −log(u)
Now substitute u back in:
−log(cos(u))
So, the result is: −2log(cos(u))
Now substitute u back in:
−2log(cos(2x))
Method #2
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The integral of a constant times a function is the constant times the integral of the function:
∫cos(2x)2sin(x)cos(x)dx=2∫cos(2x)sin(x)cos(x)dx
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Rewrite the integrand:
cos(2x)sin(x)cos(x)=2cos2(x)−1sin(x)cos(x)
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Let u=2cos2(x)−1.
Then let du=−4sin(x)cos(x)dx and substitute −4du:
∫(−4u1)du
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The integral of a constant times a function is the constant times the integral of the function:
∫u1du=−4∫u1du
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The integral of u1 is log(u).
So, the result is: −4log(u)
Now substitute u back in:
−4log(2cos2(x)−1)
So, the result is: −2log(2cos2(x)−1)
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Add the constant of integration:
−2log(cos(2x))+constant
The answer is:
−2log(cos(2x))+constant
The answer (Indefinite)
[src]
/
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| sin(2*x) log(cos(2*x))
| -------- dx = C - -------------
| cos(2*x) 2
|
/
∫cos(2x)sin(2x)dx=C−2log(cos(2x))
The graph
Use the examples entering the upper and lower limits of integration.