Integral of cos³xsin²x dx
The solution
Detail solution
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Rewrite the integrand:
sin2(x)cos3(x)=(1−sin2(x))sin2(x)cos(x)
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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:
∫(−u4+u2)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:
∫(−u4)du=−∫u4du
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The integral of un is n+1un+1 when n=−1:
∫u4du=5u5
So, the result is: −5u5
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The integral of un is n+1un+1 when n=−1:
∫u2du=3u3
The result is: −5u5+3u3
Now substitute u back in:
−5sin5(x)+3sin3(x)
Method #2
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Rewrite the integrand:
(1−sin2(x))sin2(x)cos(x)=−sin4(x)cos(x)+sin2(x)cos(x)
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Integrate term-by-term:
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−sin4(x)cos(x))dx=−∫sin4(x)cos(x)dx
-
Let u=sin(x).
Then let du=cos(x)dx and substitute du:
∫u4du
-
The integral of un is n+1un+1 when n=−1:
∫u4du=5u5
Now substitute u back in:
5sin5(x)
So, the result is: −5sin5(x)
-
Let u=sin(x).
Then let du=cos(x)dx and substitute du:
∫u2du
-
The integral of un is n+1un+1 when n=−1:
∫u2du=3u3
Now substitute u back in:
3sin3(x)
The result is: −5sin5(x)+3sin3(x)
Method #3
-
Rewrite the integrand:
(1−sin2(x))sin2(x)cos(x)=−sin4(x)cos(x)+sin2(x)cos(x)
-
Integrate term-by-term:
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−sin4(x)cos(x))dx=−∫sin4(x)cos(x)dx
-
Let u=sin(x).
Then let du=cos(x)dx and substitute du:
∫u4du
-
The integral of un is n+1un+1 when n=−1:
∫u4du=5u5
Now substitute u back in:
5sin5(x)
So, the result is: −5sin5(x)
-
Let u=sin(x).
Then let du=cos(x)dx and substitute du:
∫u2du
-
The integral of un is n+1un+1 when n=−1:
∫u2du=3u3
Now substitute u back in:
3sin3(x)
The result is: −5sin5(x)+3sin3(x)
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Add the constant of integration:
−5sin5(x)+3sin3(x)+constant
The answer is:
−5sin5(x)+3sin3(x)+constant
The answer (Indefinite)
[src]
/
| 5 3
| 3 2 sin (x) sin (x)
| cos (x)*sin (x) dx = C - ------- + -------
| 5 3
/
−153sin5x−5sin3x
The graph
5 3
sin (1) sin (1)
- ------- + -------
5 3
−153sin51−5sin31
=
5 3
sin (1) sin (1)
- ------- + -------
5 3
−5sin5(1)+3sin3(1)
Use the examples entering the upper and lower limits of integration.