$$u=3{x}^{2}-2x+1 \Rightarrow du=6x-2\ dx=2\left(3x-1\right)dx \Rightarrow \frac{1}{2}du=\left(3x-1\right)dx$$karush said:$$\int\frac{3x-1}{\left(3{x}^{2}-2x+1\right)^4} dx$$
$$u=\left(3{x}^{2}-2x+1\right) du=6x-2\ dx=2\left(3x-1\right)dx$$
got this far but when I tried to complete the ans was wrong
the correct ans is
$\frac{-1}{6\left(3x^2-2x+1 \right)^3}+C$
Integration by substitution is a method used in calculus to evaluate integrals. It involves replacing a variable in an integral with a new variable, which allows for easier integration.
Integration by substitution is useful because it can simplify the integration process for more complex functions. It allows for the integral to be written in a form that can be more easily evaluated using basic integration techniques.
The substitution variable is typically chosen based on the function being integrated. It should be a variable that, when substituted in the integral, makes the integrand simpler to work with.
The general process for integration by substitution involves selecting a substitution variable, substituting it into the integral, simplifying the integrand, and then integrating with respect to the new variable. The final step is to substitute the original variable back into the solution.
Integration by substitution is most useful for integrals with more complicated functions, such as trigonometric functions or exponential functions. It can also be used when integrals involve nested functions or when the integrand can be simplified using a substitution.