The integral ∫x2x3+1 dx by making the given substitution u=x3+1.
i) The substitution rule
If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du
ii) Indefinite integral
∫xn dx=xn+1n+1+C (n≠-1)
∫x2x3+1 dx u=x3+1
Use thesubstitution u=x3+1.
Differentiate u=x3+1 with respect to x.
As x2 dx is a part of the integration, solving for x2 dx by dividing both side by 3.
By using concept i)
Substitute u=x3+1, du3=x2dx . To write ∫x2x3+1 dx=∫u12du3
By using concept ii)