Solving a Polynomial Integral
The image contains an integral that you would like to solve. The integral is:
∫(8x^3 - x^2 + 5x - 1) dx
To solve this integral, we integrate each term separately with respect to x:
∫8x^3 dx - ∫x^2 dx + ∫5x dx - ∫1 dx
Now, for each term, apply the power rule of integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C where C is the constant of integration:
1. ∫8x^3 dx = (8/4)x^(3+1) = 2x^4
2. ∫x^2 dx = (1/3)x^(2+1) = (1/3)x^3
3. ∫5x dx = (5/2)x^(1+1) = (5/2)x^2
4. ∫1 dx = x
Now, put them all together, including the different signs:
2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
So the final answer for the indefinite integral is:
2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
