91Ó°ÊÓ

When we are tasked with finding the integral of a function, employing the right integration techniques is crucial to simplify the process and obtain the correct result. An integral is, essentially, the reverse process of differentiation and signifies the computation of the area under a curve.

In our example, the given integral is a sum of two fractions involving the variable u. Here, we use the technique of breaking down the initial complex integral into two simpler ones. Each term can then be handled individually using other integration rules and techniques. Initially intimidating expressions can usually be simplified by these methods, enabling us to use fundamental integration rules to find the antiderivative.

An antiderivative of a function is another function such that when it is differentiated, the original function is obtained. For instance, if F(u) is an antiderivative of f(u), then F'(u) = f(u). The antiderivative is not unique because adding a constant to a function does not change its derivative.

The process of finding the antiderivatives is known as integration. In our problem, once each term of the integral has been simplified, we look for their respective antiderivatives separately. Understanding antiderivatives is fundamental to mastering integration, as they represent the accumulation of quantities, such as area under the curve.

A fundamental tool in integration is the power rule, which is the direct counterpart of the power rule for differentiation. The power rule for integration is used to find the antiderivative of a power function of the form u^n. According to this rule, the integral of u^n with respect to u is given by (u^{n+1})/(n+1) plus a constant of integration, assuming that n is not equal to -1.

In the provided exercise, the power rule is applied to each term after the integral is split. For instance, the expression \(2/u^2\) becomes \(2u^{-2}\), and applying the power rule gives us \( -2u^{-1} \), which simplifies to \( -2/u \) once we adjust for the negative exponent.

The constant of integration is an essential part of the puzzle when solving indefinite integrals. Since derivatives of constants are zero, when we find the antiderivative of a function, there are an infinite number of functions that could be the original function, all differing by a constant. Hence, we add a C, the constant of integration, to account for this.

In the exercise, after finding the antiderivative of each individual component, a unique constant, C1 and C2, is added. These constants are placeholders for any real number. When combining the integrals, we can simply combine these constants into a single constant C, because the exact value is unknown unless initial conditions are given. It's a representation of the fact that multiple antiderivatives exist for any given function, differing only by a constant. Understanding this concept is key to correctly applying indefinite integration.