Integration of Polynomial Functions
To solve the given integral, we will integrate each term separately. The integral given is:
∫(8x^3 - x^2 + 5x - 1)dx
To integrate this, we use the power rule for integration:
∫x^n dx = x^(n+1) / (n+1) + C, where n ≠ -1 and C is the constant of integration.
Applying this rule for each term, we get:
∫8x^3 dx = 8 * ∫x^3 dx = 8 * (x^(3+1) / (3+1)) + C_1 = 8 * (x^4 / 4) + C_1 = 2x^4 + C_1
∫(-x^2) dx = - ∫x^2 dx = - (x^(2+1) / (2+1)) + C_2 = - (x^3 / 3) + C_2
∫5x dx = 5 * ∫x dx = 5 * (x^(1+1) / (1+1)) + C_3 = 5 * (x^2 / 2) + C_3 = (5/2)x^2 + C_3
∫(-1) dx = -x + C_4
Now, adding them all together and combining the constants of integration, we get:
∫(8x^3 - x^2 + 5x - 1)dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
Where C is the constant of integration, representing that there can be any constant added to the integral (C = C_1 + C_2 + C_3 + C_4).
