Question:

Find the indefinite integral of the function:

f(x) = 3x^2 + 4x - 5

Instructions: Show all your work and provide the answer in simplified form.

Answer: To find the indefinite integral of the function f(x), we need to use the power rule for integration. The power rule states that if we have a function of the form x^n, the indefinite integral is (1/(n+1)) * x^(n+1), except when n = -1, in which case the integral is ln|x| + C (where C is the constant of integration).

Given that f(x) = 3x^2 + 4x - 5, we can apply the power rule to each term individually.

The integral of 3x^2 will be [(1/3) * x^3] + C1, where C1 is the constant of integration for the first term.

The integral of 4x will be [(1/2) * (4x^2)] + C2, where C2 is the constant of integration for the second term.

The integral of -5 will simply be (-5x) + C3, where C3 is the constant of integration for the constant term.

Putting it all together:

∫(3x^2 + 4x - 5) dx = [(1/3) * x^3] + [(1/2) * (4x^2)] - 5x + C

Simplifying further, we get:

∫(3x^2 + 4x - 5) dx = (x^3)/3 + (2x^2) - 5x + C

Therefore, the indefinite integral of f(x) = 3x^2 + 4x - 5 is (x^3)/3 + (2x^2) - 5x + C, where C is the constant of integration.