Integration of a Polynomial Function
Sure, the given integral is:
∫ (8x^3 - x^2 + 5x - 1) dx
We need to integrate each term separately. The integral of a polynomial function is found by increasing the exponent by one and dividing by the new exponent. Here's how it works for each term:
1. Integral of 8x^3 dx: Increase exponent by 1 (from 3 to 4), then divide by the new exponent (4).
∫ 8x^3 dx = 8/4 x^4 = 2x^4
2. Integral of -x^2 dx: Increase exponent by 1 (from 2 to 3), then divide by the new exponent (3).
∫ -x^2 dx = -1/3 x^3
3. Integral of 5x dx: Increase exponent by 1 (from 1 to 2), then divide by the new exponent (2).
∫ 5x dx = 5/2 x^2
4. Integral of -1 dx: Since the exponent is 0 (because -1 is the same as -1x^0), we just multiply x by the constant.
∫ -1 dx = -1 * x = -x
Now, let's put it all together:
∫ (8x^3 - x^2 + 5x - 1) dx = 2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
Don't forget to add the constant of integration, C, at the end.
The final answer is:
2x^4 - (1/3)x^3 + (5/2)x^2 - x + C
