Integral of cos(2pix)cos(pix) 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=πx.
Then let du=πdx and substitute πdu:
∫π2cos(u)cos(2u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫πcos(2u)cos(u)du=π∫cos(2u)cos(u)du
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Rewrite the integrand:
cos(2u)cos(u)=2cos3(u)−cos(u)
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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:
∫2cos3(u)du=2∫cos3(u)du
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Rewrite the integrand:
cos3(u)=(1−sin2(u))cos(u)
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Let u=sin(u).
Then let du=cos(u)du and substitute du:
∫(1−u2)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:
∫1du=u
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The integral of a constant times a function is the constant times the integral of the function:
∫(−u2)du=−∫u2du
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The integral of un is n+1un+1 when n=−1:
∫u2du=3u3
So, the result is: −3u3
The result is: −3u3+u
Now substitute u back in:
−3sin3(u)+sin(u)
So, the result is: −32sin3(u)+2sin(u)
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The integral of a constant times a function is the constant times the integral of the function:
∫(−cos(u))du=−∫cos(u)du
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The integral of cosine is sine:
∫cos(u)du=sin(u)
So, the result is: −sin(u)
The result is: −32sin3(u)+sin(u)
So, the result is: π−32sin3(u)+sin(u)
Now substitute u back in:
π−32sin3(πx)+sin(πx)
Method #2
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Rewrite the integrand:
cos(πx)cos(2πx)=2cos3(πx)−cos(πx)
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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:
∫2cos3(πx)dx=2∫cos3(πx)dx
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Rewrite the integrand:
cos3(πx)=(1−sin2(πx))cos(πx)
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Let u=sin(πx).
Then let du=πcos(πx)dx and substitute πdu:
∫π21−u2du
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The integral of a constant times a function is the constant times the integral of the function:
∫π1−u2du=π∫(1−u2)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:
∫1du=u
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−u2)du=−∫u2du
-
The integral of un is n+1un+1 when n=−1:
∫u2du=3u3
So, the result is: −3u3
The result is: −3u3+u
So, the result is: π−3u3+u
Now substitute u back in:
π−3sin3(πx)+sin(πx)
So, the result is: π2(−3sin3(πx)+sin(πx))
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The integral of a constant times a function is the constant times the integral of the function:
∫(−cos(πx))dx=−∫cos(πx)dx
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Let u=πx.
Then let du=πdx and substitute πdu:
∫π2cos(u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫πcos(u)du=π∫cos(u)du
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The integral of cosine is sine:
∫cos(u)du=sin(u)
So, the result is: πsin(u)
Now substitute u back in:
πsin(πx)
So, the result is: −πsin(πx)
The result is: π2(−3sin3(πx)+sin(πx))−πsin(πx)
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Now simplify:
3π(cos(2πx)+2)sin(πx)
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Add the constant of integration:
3π(cos(2πx)+2)sin(πx)+constant
The answer is:
3π(cos(2πx)+2)sin(πx)+constant
The answer (Indefinite)
[src]
3
2*sin (pi*x)
/ - ------------ + sin(pi*x)
| 3
| cos(2*pi*x)*cos(pi*x) dx = C + --------------------------
| pi
/
6πsin(3πx)+2πsin(πx)
The graph
6πsin(3π)+3sinπ
=
Use the examples entering the upper and lower limits of integration.