The indefinite integral is a fundamental theorem in all calculus courses. They are also referred to as antiderivatives because the process of calculating a derivative is reversed to find the integral. For this example, f(x) = x^3 is used as a guide.
Continue ReadingThe first step to find the integral of the function f(x) = x^3 is to add one digit to the exponent. The result is: f(x) = x^(3+1) or f(x) = x^4.
After finding the new exponent, the next step is to divide by it. Because 1 is believed to be in front of the "x" in the function, the result is: f(x) = (1/4)x^4.
After the first two steps, the integral is nearly complete. The only thing missing is a constant that cannot be accounted for by just simply integrating. Therefore, mathematicians add a "C" to the end of the function. The final integral is: f(x) = (1/4)x^4 + C.
In order to prove the integral is correct, simply derive it. The result is f(x) = (4/4) x^(4-1) or the original function of f(x) = x^3.