Integral of x³lnxdx dx
The solution
Detail solution
-
There are multiple ways to do this integral.
Method #1
-
Let u=log(x).
Then let du=xdx and substitute du:
∫ue4udu
-
Use integration by parts:
∫udv=uv−∫vdu
Let u(u)=u and let dv(u)=e4u.
Then du(u)=1.
To find v(u):
-
There are multiple ways to do this integral.
Method #1
-
Let u=4u.
Then let du=4du and substitute 4du:
∫16eudu
-
The integral of a constant times a function is the constant times the integral of the function:
∫4eudu=4∫eudu
-
The integral of the exponential function is itself.
∫eudu=eu
So, the result is: 4eu
Now substitute u back in:
4e4u
Method #2
-
Let u=e4u.
Then let du=4e4udu and substitute 4du:
∫161du
-
The integral of a constant times a function is the constant times the integral of the function:
∫41du=4∫1du
-
The integral of a constant is the constant times the variable of integration:
∫1du=u
So, the result is: 4u
Now substitute u back in:
4e4u
Now evaluate the sub-integral.
-
The integral of a constant times a function is the constant times the integral of the function:
∫4e4udu=4∫e4udu
-
Let u=4u.
Then let du=4du and substitute 4du:
∫16eudu
-
The integral of a constant times a function is the constant times the integral of the function:
∫4eudu=4∫eudu
-
The integral of the exponential function is itself.
∫eudu=eu
So, the result is: 4eu
Now substitute u back in:
4e4u
So, the result is: 16e4u
Now substitute u back in:
4x4log(x)−16x4
Method #2
-
Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=log(x) and let dv(x)=x3.
Then du(x)=x1.
To find v(x):
-
The integral of xn is n+1xn+1 when n=−1:
∫x3dx=4x4
Now evaluate the sub-integral.
-
The integral of a constant times a function is the constant times the integral of the function:
∫4x3dx=4∫x3dx
-
The integral of xn is n+1xn+1 when n=−1:
∫x3dx=4x4
So, the result is: 16x4
-
Now simplify:
16x4⋅(4log(x)−1)
-
Add the constant of integration:
16x4⋅(4log(x)−1)+constant
The answer is:
16x4⋅(4log(x)−1)+constant
The answer (Indefinite)
[src]
/
| 4 4
| 3 x x *log(x)
| x *log(x)*1 dx = C - -- + ---------
| 16 4
/
4x4logx−16x4
Use the examples entering the upper and lower limits of integration.