Solución
The input is rewritten: ∫ln(x2)dx=∫2ln(x)dx.
Apply the constant multiple rule ∫cf(x)dx=c∫f(x)dx with c=2 and f(x)=ln(x):
∫2ln(x)dx=(2∫ln(x)dx)
For the integral ∫ln(x)dx, use integration by parts ∫udv=uv−∫vdu.
Let u=ln(x) and dv=dx.
Then du=(ln(x))′dx=xdx (steps can be seen ») and v=∫1dx=x (steps can be seen »).
The integral becomes
2∫ln(x)dx=2(ln(x)⋅x−∫x⋅x1dx)=2(xln(x)−∫1dx)
Apply the constant rule ∫cdx=cx with c=1:
2xln(x)−2∫1dx=2xln(x)−2x
Por lo tanto,
∫2ln(x)dx=2xln(x)−2x
Simplifica:
∫2ln(x)dx=2x(ln(x)−1)
Añade la constante de integración:
∫2ln(x)dx=2x(ln(x)−1)+C
Answer: ∫2ln(x)dx=2x(ln(x)−1)+C