Integral of cos^3(4x) dx
The solution
Detail solution
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Rewrite the integrand:
cos3(4x)=(1−sin2(4x))cos(4x)
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There are multiple ways to do this integral.
Method #1
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Let u=sin(4x).
Then let du=4cos(4x)dx and substitute du:
∫(41−4u2)du
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Integrate term-by-term:
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The integral of a constant is the constant times the variable of integration:
∫41du=4u
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The integral of a constant times a function is the constant times the integral of the function:
∫(−4u2)du=−4∫u2du
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The integral of un is n+1un+1 when n=−1:
∫u2du=3u3
So, the result is: −12u3
The result is: −12u3+4u
Now substitute u back in:
−12sin3(4x)+4sin(4x)
Method #2
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Rewrite the integrand:
(1−sin2(4x))cos(4x)=−sin2(4x)cos(4x)+cos(4x)
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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:
∫(−sin2(4x)cos(4x))dx=−∫sin2(4x)cos(4x)dx
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Let u=sin(4x).
Then let du=4cos(4x)dx and substitute 4du:
∫16u2du
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The integral of a constant times a function is the constant times the integral of the function:
∫4u2du=4∫u2du
-
The integral of un is n+1un+1 when n=−1:
∫u2du=3u3
So, the result is: 12u3
Now substitute u back in:
12sin3(4x)
So, the result is: −12sin3(4x)
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Let u=4x.
Then let du=4dx and substitute 4du:
∫16cos(u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫4cos(u)du=4∫cos(u)du
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The integral of cosine is sine:
∫cos(u)du=sin(u)
So, the result is: 4sin(u)
Now substitute u back in:
4sin(4x)
The result is: −12sin3(4x)+4sin(4x)
Method #3
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Rewrite the integrand:
(1−sin2(4x))cos(4x)=−sin2(4x)cos(4x)+cos(4x)
-
Integrate term-by-term:
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−sin2(4x)cos(4x))dx=−∫sin2(4x)cos(4x)dx
-
Let u=sin(4x).
Then let du=4cos(4x)dx and substitute 4du:
∫16u2du
-
The integral of a constant times a function is the constant times the integral of the function:
∫4u2du=4∫u2du
-
The integral of un is n+1un+1 when n=−1:
∫u2du=3u3
So, the result is: 12u3
Now substitute u back in:
12sin3(4x)
So, the result is: −12sin3(4x)
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Let u=4x.
Then let du=4dx and substitute 4du:
∫16cos(u)du
-
The integral of a constant times a function is the constant times the integral of the function:
∫4cos(u)du=4∫cos(u)du
-
The integral of cosine is sine:
∫cos(u)du=sin(u)
So, the result is: 4sin(u)
Now substitute u back in:
4sin(4x)
The result is: −12sin3(4x)+4sin(4x)
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Now simplify:
12(3−sin2(4x))sin(4x)
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Add the constant of integration:
12(3−sin2(4x))sin(4x)+constant
The answer is:
12(3−sin2(4x))sin(4x)+constant
The answer (Indefinite)
[src]
/
| 3
| 3 sin (4*x) sin(4*x)
| cos (4*x) dx = C - --------- + --------
| 12 4
/
4sin(4x)−3sin3(4x)
The graph
3
sin (4) sin(4)
- ------- + ------
12 4
−12sin34−3sin4
=
3
sin (4) sin(4)
- ------- + ------
12 4
4sin(4)−12sin3(4)
Use the examples entering the upper and lower limits of integration.