Integral of cosxsin3x dx
The solution
Detail solution
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Rewrite the integrand:
sin(3x)cos(x)=−4sin3(x)cos(x)+3sin(x)cos(x)
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Integrate term-by-term:
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The integral of a constant times a function is the constant times the integral of the function:
∫(−4sin3(x)cos(x))dx=−4∫sin3(x)cos(x)dx
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There are multiple ways to do this integral.
Method #1
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Let u=sin(x).
Then let du=cos(x)dx and substitute du:
∫u3du
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The integral of un is n+1un+1 when n=−1:
∫u3du=4u4
Now substitute u back in:
4sin4(x)
Method #2
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Rewrite the integrand:
sin3(x)cos(x)=(1−cos2(x))sin(x)cos(x)
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Let u=−cos2(x).
Then let du=2sin(x)cos(x)dx and substitute du:
∫(2u+21)du
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Integrate term-by-term:
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The integral of a constant times a function is the constant times the integral of the function:
∫2udu=2∫udu
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The integral of un is n+1un+1 when n=−1:
∫udu=2u2
So, the result is: 4u2
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The integral of a constant is the constant times the variable of integration:
∫21du=2u
The result is: 4u2+2u
Now substitute u back in:
4cos4(x)−2cos2(x)
So, the result is: −sin4(x)
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The integral of a constant times a function is the constant times the integral of the function:
∫3sin(x)cos(x)dx=3∫sin(x)cos(x)dx
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Let u=cos(x).
Then let du=−sin(x)dx 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(x)
So, the result is: −23cos2(x)
The result is: −sin4(x)−23cos2(x)
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Add the constant of integration:
−sin4(x)−23cos2(x)+constant
The answer is:
−sin4(x)−23cos2(x)+constant
The answer (Indefinite)
[src]
/ 2
| 4 3*cos (x)
| cos(x)*sin(3*x) dx = C - sin (x) - ---------
| 2
/
∫sin(3x)cos(x)dx=C−sin4(x)−23cos2(x)
The graph
3 3*cos(1)*cos(3) sin(1)*sin(3)
- - --------------- - -------------
8 8 8
−8sin(1)sin(3)−83cos(1)cos(3)+83
=
3 3*cos(1)*cos(3) sin(1)*sin(3)
- - --------------- - -------------
8 8 8
−8sin(1)sin(3)−83cos(1)cos(3)+83
3/8 - 3*cos(1)*cos(3)/8 - sin(1)*sin(3)/8
Use the examples entering the upper and lower limits of integration.