tmt said:I have this integral
$$6\int_{}^{} \frac{u^3 - 1 + 1}{u - 1}\,d$$
And I need to simplify it to
$$6\int_{}^{}u^2 + u + 1 + \frac{1}{u - 1}\,du$$
But I don't know how to get to this step.
An integral is a mathematical concept that represents the area under a curve. It is used in calculus to find the value of a function over a certain interval.
Simplifying integrals makes them easier to solve and allows us to find the exact value of the integral without having to use complex methods. It also helps us to better understand the behavior of the function.
The steps to simplify an integral include using the properties of integrals, such as linearity and power rule, to rewrite the integral in a simpler form. Then, you can use substitution or integration by parts to further simplify the integral.
Substitution is useful when the integral contains a complicated expression or when the limits of integration are difficult to work with. It allows us to replace the variable in the integral with a new variable, making it easier to solve.
Integration by parts is a method used to simplify integrals that involve the product of two functions. It involves using the product rule of derivatives to rewrite the integral in a simpler form. This method is useful when the integral cannot be solved by other methods, such as substitution or using properties of integrals.