Integral of Polynomial Function
The integral in the image is an indefinite integral of a polynomial function. To solve the integral, you would integrate each term separately, applying the power rule for integration. Here's the integration term-by-term:
∫(8x^3 - x^2 + 5x - 1) dx
= ∫8x^3 dx - ∫x^2 dx + ∫5x dx - ∫1 dx
Now applying the power rule for integration (which states that ∫x^n dx = x^(n+1)/(n+1) for all n ≠ -1), we get:
= 8 * x^(3+1)/(3+1) - x^(2+1)/(2+1) + 5 * x^(1+1)/(1+1) - x + C
= 8 * x^4/4 - x^3/3 + 5 * x^2/2 - x + C
Simplifying, we have:
= 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
So the integral is:
2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
where C represents the constant of integration.
