LCKurtz said:The familiar antiderivative formula
[tex]\int x^n\, dx = \frac{x^{n+1}}{n+1}+C[/tex]
also works for negative exponents. Write your fraction as a negative exponent.
discy said:thanks for your answer.
now how would I put 1/x³ into that formula to get -0.5 * x^-2 ?
An anti-derivative is the inverse operation of a derivative. It is a mathematical function that, when differentiated, gives the original function.
Computing the anti-derivative of a function allows us to find the original function from its derivative. In this case, 1/x^3 is a commonly used function in physics and engineering, so knowing its anti-derivative can help solve many problems.
The general method for computing the anti-derivative of 1/x^3 is using the power rule for integration, which states that the anti-derivative of x^n is (x^(n+1))/(n+1). In this case, we would use this rule with n=-3, giving us the anti-derivative of 1/x^3 as -1/(2x^2).
Yes, there are some special cases when computing the anti-derivative of 1/x^3. For example, if the function also has a constant added to it, we would need to use the power rule along with the constant rule for integration. Additionally, if the function is in a different form, such as (ax+b)^n, we would need to use the substitution method to find the anti-derivative.
One way to check if your computed anti-derivative is correct is by taking the derivative of the anti-derivative and seeing if it gives you the original function. In the case of 1/x^3, taking the derivative of -1/(2x^2) would give you 1/x^3, confirming that your computed anti-derivative is correct.