Mastering the Constant Multiple Rule for Evaluating Integrals – A Comprehensive Guide

∫kf(u) du

To evaluate the integral ∫kf(u) du, we can use the constant multiple rule of integration

To evaluate the integral ∫kf(u) du, we can use the constant multiple rule of integration. According to the constant multiple rule, if k is a constant, we can take it outside the integral:

∫kf(u) du = k ∫f(u) du

Now, we need to evaluate the integral of f(u) with respect to u. The specific method of integration would depend on the function f(u) itself. If f(u) is a simple function, it may have a known antiderivative, in which case we can directly find the solution. If f(u) is a more complex function, we may need to use integration techniques such as substitution, integration by parts, or partial fraction decomposition.

Let’s consider an example to illustrate this concept. Suppose we have the integral ∫3(2u + 1) du. We can apply the constant multiple rule:

∫3(2u + 1) du = 3 ∫2u + 1 du

Now, we separately integrate each term:

∫2u du + ∫1 du = 3 ∫2u du + 3 ∫1 du

Integrating each term individually, we get:

u^2 + u + C

So, the final result is:

∫3(2u + 1) du = u^2 + u + C

It’s important to remember to include the constant of integration (C) when evaluating indefinite integrals.

In summary, to evaluate the integral ∫kf(u) du, we use the constant multiple rule to take the constant (k) outside the integral and then integrate f(u) by applying the appropriate integration techniques depending on the form of f(u).

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