"Evaluate the following indefinite integrals:
"S" (3x^2 -2)/(x^3 - 2x + 1)^3 dx"
We're practicing the substitution rule, and I know how to do it, but I don't know what/how to substitute in this question.
btw: "S" is the integral sign.
substituition for the denominator
giving you du= 3x^2-2
leaving you with just 1/u^3 then find the anti derivative of that and youll have you answer. Don't forget your c
To evaluate the given indefinite integral ∫ (3x^2 - 2) / (x^3 - 2x + 1)^3 dx, you can indeed make use of the substitution rule. To determine the appropriate substitution, we need to identify a part of the integrand that resembles the derivative of an inside function.
Let's start by looking at the denominator, (x^3 - 2x + 1)^3. Notice that the derivative of (x^3 - 2x + 1) is 3x^2 - 2. Therefore, we can choose to substitute u = x^3 - 2x + 1. Taking the derivative of u with respect to x yields du = (3x^2 - 2) dx.
To proceed, let's rewrite the original integral with the new variable: ∫ (3x^2 - 2) / (x^3 - 2x + 1)^3 dx = ∫ du / u^3.
Now, our integral becomes ∫ du / u^3, which can be evaluated more easily. Using the power rule for integration, we have ∫ du / u^3 = -1 / (2u^2) + C, where C represents the constant of integration.
Finally, substituting back u = x^3 - 2x + 1 into the result, we get the final answer as -1 / (2(x^3 - 2x + 1)^2) + C.