Integral of x^2*sin(2x) dx
The solution
Detail solution
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There are multiple ways to do this integral.
Method #1
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=x2 and let dv(x)=sin(2x).
Then du(x)=2x.
To find v(x):
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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:
∫4sin(u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫2sin(u)du=2∫sin(u)du
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The integral of sine is negative cosine:
∫sin(u)du=−cos(u)
So, the result is: −2cos(u)
Now substitute u back in:
−2cos(2x)
Method #2
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The integral of a constant times a function is the constant times the integral of the function:
∫2sin(x)cos(x)dx=2∫sin(x)cos(x)dx
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Let u=cos(x).
Then let du=−sin(x)dx and substitute −du:
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The integral of a constant times a function is the constant times the integral of the function:
∫(−u)du=−∫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: −cos2(x)
Now evaluate the sub-integral.
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=−x and let dv(x)=cos(2x).
Then du(x)=−1.
To find v(x):
-
Let u=2x.
Then let du=2dx and substitute 2du:
∫4cos(u)du
-
The integral of a constant times a function is the constant times the integral of the function:
∫2cos(u)du=2∫cos(u)du
-
The integral of cosine is sine:
∫cos(u)du=sin(u)
So, the result is: 2sin(u)
Now substitute u back in:
2sin(2x)
Now evaluate the sub-integral.
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The integral of a constant times a function is the constant times the integral of the function:
∫(−2sin(2x))dx=−2∫sin(2x)dx
-
Let u=2x.
Then let du=2dx and substitute 2du:
∫4sin(u)du
-
The integral of a constant times a function is the constant times the integral of the function:
∫2sin(u)du=2∫sin(u)du
-
The integral of sine is negative cosine:
∫sin(u)du=−cos(u)
So, the result is: −2cos(u)
Now substitute u back in:
−2cos(2x)
So, the result is: 4cos(2x)
Method #2
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The integral of a constant times a function is the constant times the integral of the function:
∫2x2sin(x)cos(x)dx=2∫x2sin(x)cos(x)dx
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=x2 and let dv(x)=sin(x)cos(x).
Then du(x)=2x.
To find v(x):
-
Let u=cos(x).
Then let du=−sin(x)dx and substitute −du:
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−u)du=−∫udu
-
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)
Now evaluate the sub-integral.
-
Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=−x and let dv(x)=cos2(x).
Then du(x)=−1.
To find v(x):
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Rewrite the integrand:
cos2(x)=2cos(2x)+21
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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:
∫2cos(2x)dx=2∫cos(2x)dx
-
Let u=2x.
Then let du=2dx and substitute 2du:
∫4cos(u)du
-
The integral of a constant times a function is the constant times the integral of the function:
∫2cos(u)du=2∫cos(u)du
-
The integral of cosine is sine:
∫cos(u)du=sin(u)
So, the result is: 2sin(u)
Now substitute u back in:
2sin(2x)
So, the result is: 4sin(2x)
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The integral of a constant is the constant times the variable of integration:
∫21dx=2x
The result is: 2x+4sin(2x)
Now evaluate the sub-integral.
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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:
∫(−2x)dx=−2∫xdx
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The integral of xn is n+1xn+1 when n=−1:
∫xdx=2x2
So, the result is: −4x2
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The integral of a constant times a function is the constant times the integral of the function:
∫(−4sin(2x))dx=−4∫sin(2x)dx
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Let u=2x.
Then let du=2dx and substitute 2du:
∫4sin(u)du
-
The integral of a constant times a function is the constant times the integral of the function:
∫2sin(u)du=2∫sin(u)du
-
The integral of sine is negative cosine:
∫sin(u)du=−cos(u)
So, the result is: −2cos(u)
Now substitute u back in:
−2cos(2x)
So, the result is: 8cos(2x)
The result is: −4x2+8cos(2x)
So, the result is: −x2cos2(x)−2x2+2x(2x+4sin(2x))+4cos(2x)
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Add the constant of integration:
−2x2cos(2x)+2xsin(2x)+4cos(2x)+constant
The answer is:
−2x2cos(2x)+2xsin(2x)+4cos(2x)+constant
The answer (Indefinite)
[src]
/
| 2
| 2 cos(2*x) x*sin(2*x) x *cos(2*x)
| x *sin(2*x) dx = C + -------- + ---------- - -----------
| 4 2 2
/
84xsin(2x)+(2−4x2)cos(2x)
The graph
1 sin(2) cos(2)
- - + ------ - ------
4 2 4
42sin2−cos2−41
=
1 sin(2) cos(2)
- - + ------ - ------
4 2 4
−41−4cos(2)+2sin(2)
Use the examples entering the upper and lower limits of integration.