Integral of x*asin(x^2) 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=x2.
Then let du=2xdx and substitute 2du:
∫2asin(u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫asin(u)du=2∫asin(u)du
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Use integration by parts:
∫udv=uv−∫vdu
Let u(u)=asin(u) and let dv(u)=1.
Then du(u)=1−u21.
To find v(u):
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The integral of a constant is the constant times the variable of integration:
∫1du=u
Now evaluate the sub-integral.
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Let u=1−u2.
Then let du=−2udu and substitute −2du:
∫(−2u1)du
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The integral of a constant times a function is the constant times the integral of the function:
∫u1du=−2∫u1du
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The integral of un is n+1un+1 when n=−1:
∫u1du=2u
So, the result is: −u
Now substitute u back in:
−1−u2
So, the result is: 2uasin(u)+21−u2
Now substitute u back in:
2x2asin(x2)+21−x4
Method #2
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=asin(x2) and let dv(x)=x.
Then du(x)=1−x42x.
To find v(x):
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The integral of xn is n+1xn+1 when n=−1:
∫xdx=2x2
Now evaluate the sub-integral.
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Let u=1−x4.
Then let du=−4x3dx and substitute −4du:
∫(−4u1)du
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The integral of a constant times a function is the constant times the integral of the function:
∫u1du=−4∫u1du
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The integral of un is n+1un+1 when n=−1:
∫u1du=2u
So, the result is: −2u
Now substitute u back in:
−21−x4
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Add the constant of integration:
2x2asin(x2)+21−x4+constant
The answer is:
2x2asin(x2)+21−x4+constant
The answer (Indefinite)
[src]
/ ________
| / 4 2 / 2\
| / 2\ \/ 1 - x x *asin\x /
| x*asin\x / dx = C + ----------- + -----------
| 2 2
/
∫xasin(x2)dx=C+2x2asin(x2)+21−x4
The graph
−21+4π
=
−21+4π
Use the examples entering the upper and lower limits of integration.