The integral ∫x3x4-52 dx by making the given substitution u=x4-5.
i) The substitution rule is that
If u=g(x) is a differentiable function whose range is an interval I and f is continuous oninterval I, then ∫f(gx)g'xdx=∫f(u)du.
ii) Indefinite integral
∫xn dx=xn+1n+1+C (n≠-1)
∫x3x4-52 dx u=x4-5
Use the substitution u=x4-5
Differentiate u=(x4-5) with respect to x
As x3 dx is a part of the integration, solving for x3 dx by dividing both side by 4.
By using concept i)
Substitute u=x4-5,x3dx=du4 to get ∫x3x4-52 dx=∫1u2du4
By using concept ii)