R
h
x−x
3
x
3
i
dx
Using the power rule of integration, we can integrate (x^{-1/3}) dx:
∫(x^{-1/3})dx = (x^{2/3})/(2/3) + C
Simplifying, we get:
∫(x^{-1/3})dx = (3/2)x^{2/3} + C
Therefore, the result of the integral ∫(h(x−x^{1/3}))dx is:
∫(h(x - x^{1/3}))dx = h(3/2)x^{2/3} - h(2/5)x^{5/3} + C
To simplify the expression ∫(h * (x - x^3) * i) dx step by step, we can break it down into smaller parts:
Step 1: Distribute the variables h and i into the expression (x - x^3):
∫(h * (x - x^3) * i) dx = ∫(hxi - hi(x^3)) dx
Step 2: Integrate each term separately:
∫(hxi - hi(x^3)) dx = ∫hxi dx - ∫hi(x^3) dx
Step 3: Integrate each term using the power rule of integration:
∫hxi dx = h * (1/2)x^2 + C1
∫hi(x^3) dx = hi * (1/4)x^4 + C2
where C1 and C2 are constants of integration.
Therefore, the final result of the expression ∫(h * (x - x^3) * i) dx is:
∫(h * (x - x^3) * i) dx = h * (1/2)x^2 - hi * (1/4)x^4 + C, where C is the constant of integration.