Question

Complete the table by identifying \(u\) and \(d u\) for the integral. $$ \int f(g(x)) g^{\prime}(x) d x \quad \underline{u=g(x)} \quad \underline{d u=g^{\prime}(x) d x} $$ $$ \int\left(5 x^{2}+1\right)^{2}(10 x) d x $$

Short answer

In the given integral, \(u\) is \((5x^2+1)\) and \(du\) is \(10x dx\).

Step-by-step solution

Identifying \(u=g(x)\)

When selecting \(u\), we look for a function that its derivative, or a constant multiple thereof, is also included in the given integral. In this exercise, we can see that the function \((5x^2+1)\) is within a power, 2, and its derivative, \(10x\), appears in the integral. So we say \(u=(5x^2+1)\).

Identifying \(d u = g^{\prime}(x) d x\)

To identify \(du\), we have to find the derivative of \(u\) with respect to \(x\). The derivative of \(u=(5x^2+1)\) with respect to \(x\) is \(10x\). Therefore, \(du = 10x dx\).

Key concepts

U-Substitution Method

When you encounter an integration problem that seems complex due to a function within a function, the u-substitution method is a powerful tool that simplifies the process. It’s akin to unwinding a tightly rolled thread by pulling the right end. The method involves substituting part of the integral with a new variable, typically denoted as 'u'. The goal is to rewrite the integral in terms of 'u', making it easier to solve.

Consider the integral of a composed function, like in the exercise \( \int f(g(x)) g'(x) dx \) - it's tricky because you have the derivative of a function multiplied by the function itself. Here, the u-substitution method comes into play: you let \( u = g(x) \) and then express \( du \) in terms of \( dx \) as \( du = g'(x) dx \), transforming the integral into \( \int f(u) du \) which is usually much simpler to integrate.\

In the provided exercise, \( u \) is defined as \( (5x^2+1) \) with \( du \) being \( 10x dx \) after differentiation. By substituting these into the original integral, the problem becomes easier to tackle as you work with \( u \) instead of \( g(x) \).

Here's the step-up on simplicity: Instead of wrestling with the original integral, you now have a more straightforward expression to integrate, \( \int u^2 du \), which is a standard power rule integral. After you perform the integration with respect to 'u', don’t forget to substitute back the original \( g(x) \) terms to complete the problem.

Indefinite Integral

An indefinite integral, also known as an antiderivative, represents a family of functions whose derivative is the given function. In simpler terms, integrating is the opposite of differentiating. If differentiation gives you the rate at which something changes, then integration gives you the quantity that accumulates as a result of that rate.

For example, if you differentiate \( x^2 \) you get \( 2x \) - that's the rate at which \( x^2 \) is changing. But when you integrate \( 2x \), you're essentially asking, 'What do I need to differentiate to get \( 2x \)?' And the answer leads you back to \( x^2 \) plus a constant of integration, symbolized as 'C', which accounts for all possible constants that disappeared during differentiation.

One key point in understanding indefinite integrals is that the integration process is not always straightforward. There isn’t always a direct formula to use, and sometimes, like in our exercise \( \int (5x^2+1)^2 (10x) dx \), you need to manipulate the integral to make it work—the role u-substitution often plays. This combination of integration techniques molds the skillset needed for solving complex integrals.

Chain Rule Derivative

The chain rule is one of the fundamental tools of calculus, used when taking the derivative of composite functions. Imagine a set of gears interlocked where turning one gear causes the next gear to turn. The chain rule handles such interactions in differentiation. It states that to find the derivative of a composite function \( f(g(x)) \) you multiply the derivative of the outer function, \( f' \) evaluated at \( g(x) \) by the derivative of the inner function, \( g' \).

In terms of an equation, the chain rule can be written as \( \frac{d}{dx}[f(g(x))] = f'(g(x))*g'(x) \). Applying this to our exercise, \( u = g(x) = 5x^2+1 \) is the inner function, and \( u^2 \) would be part of the outer function, with the derivative \( g'(x) = 10x \) to give us \( du = 10x dx \).

The beauty of the chain rule in integration problems is that, when recognized, it often indicates a path for u-substitution, as we have seen in the exercise provided. The derivative \( g'(x) \) forms part of the integral, signaling that u-substitution might greatly simplify your task when finding the indefinite integral.