Question

True or False By \(u\) -substitution, \(\int_{0}^{\pi / 4} \tan ^{3} x \sec ^{2} x d x=\) \(\int_{0}^{\pi / 4} u^{3} d u .\) Justify your answer.

Short answer

True. By \(u\)-substitution, \( \int_{0}^{\pi / 4} \tan ^{3} x \sec ^{2} x d x = \int_{0}^{\pi / 4} u^{3} d u \).

Step-by-step solution

Understand the \(u\)-substitution Rule

The \(u\)-substitution rule in integration involves replacing a function \(u=f(x)\) within the given integral, with its derivative \(f'(x) dx = du\), in order to simplify the integral.

Identify the function and its derivative

The given integral is \(\int_{0}^{\pi / 4} \tan ^{3} x \sec ^{2} x d x\). We identify \( \tan{x} \) as the function. Its derivative is \( \sec^2{x} \). This fits the \(u\)-substitution model since we see both the function (\( \tan{x} \)) raised to some power and its derivative (\( \sec^2{x} \)) in the integral.

Apply \(u\)-substitution

Let \(u = \tan{x}\). Then \(du = \sec^2{x} dx\). We substitute these into the integral to obtain \(\int u^3 du\). Thus, by \(u\)-substitution, the given integral does indeed become \(\int_{0}^{\pi / 4} u^{3} d u\).

Key concepts

Integral Calculus

Integral calculus is a branch of mathematical analysis that deals with the integration of functions and the finding of a function's antiderivative. At the heart of integral calculus is the concept of finding the area under a curve or the accumulation of quantities.
In many cases, the process of integration is straightforward, however, there are functions that are difficult to integrate directly. This is where integration techniques, like the method of u-substitution, come into play. The goal is to transform a complex integral into a simpler one that we can evaluate more easily.
For example, let's consider the function \(\tan ^{3} x \(sec ^{2} x \)dx\). To integrate this directly would be quite challenging due to the powers of trigonometric functions. However, by noticing a function and its derivative within the integral, we can apply the u-substitution technique to simplify the problem - which is the stepping stone in mastering integral calculus.

Integration Techniques

Integration techniques are various methods used to solve integrals that are not readily solvable using basic integration rules. One of the most common techniques is u-substitution, which we've seen in the example problem.

Understanding U-Substitution

U-substitution is often used when an integral contains a function and its derivative. By substituting the function with a single variable \(u\), we can simplify the integral. In the given exercise, we identify \(\tan x\) as the function to substitute. Its derivative, \(\sec^2 x\), is present in the integral, confirming that u-substitution is suitable.
After substitution, the integral \(\int_{0}^{\pi / 4} \tan^{3} x \sec^{2} x dx\) is transformed to \(\int u^3 du\), a much simpler integral to solve. This technique not only streamlines the solving process but also improves our understanding of how different functions relate to each other, especially in the context of their derivatives.

Definite Integrals

Definite integrals refer to the integral of a function over a specific interval. The notation \(\int_{a}^{b} f(x) dx\) represents the definite integral of the function \(f(x)\) from \(x = a\) to \(x = b\). These integrals play a critical role in applications such as computing the area under a curve, the total accumulation of a quantity, or the average value of a function on an interval.
When we apply u-substitution to a definite integral, it's essential to adjust the limits of integration to match the new variable \(u\). However, in some educational contexts, the focus is on practicing the technique rather than evaluating the definite integral. As is the case in our exercise, where the limits stay in terms of \(x\) but could be changed to terms of \(u\) if we were to evaluate the integral completely.
The ability to navigate through definite integrals is crucial in integral calculus, as it solidifies our understanding of the practical implications of integration and its power to describe physical phenomena.