Example Question - integral computation

Solving Integral of Polynomial Function

The image shows an integral that needs to be solved. The expression within the integral is a polynomial function: ∫(8x^3 - x^2 + 5x - 1) dx To solve this, we integrate each term of the polynomial separately with respect to x. The integral of a constant times a power of x is the constant times the fraction of the incremented power of x divided by the new exponent. The integral of a constant is just the constant times x. Here's how this integral is computed: ∫8x^3 dx = 2x^4 (because you add 1 to the exponent and then divide by the new exponent) ∫-x^2 dx = -x^3/3 (same principle) ∫5x dx = 5x^2/2 (again, same principle) ∫-1 dx = -x Putting it all together, the antiderivative of the given function is: 2x^4 - x^3/3 + 5x^2/2 - x + C where C is the constant of integration.