Question

Different Solutions? Consider the integral \(\int 2 \sin x \cos x d x\) (a) Evaluate the integral using the substitution \(u=\sin x\) (b) Evaluate the integral using the substitution \(u=\cos x\) . (c) Writing to Learn Explain why the different-looking answers in parts (a) and (b) are actually equivalent.

Short answer

The integral \(\int 2 \sin x \cos x d x\) evaluated using the substitution \(u=\sin x\) yields \(\sin^2 x + C\), and using the substitution \(u= \cos x\) yields \(-\cos^2 x + C\). These results may look different but are equivalent due to the Pythagorean Identity in trigonometry \(\sin^2 x + \cos^2 x = 1\).

Step-by-step solution

Solving integral with the substitution \(u=\sin x\)

The given integral is \(\int 2 \sin x \cos x d x\). Substituting \(u=\sin x\), we get \(du=\cos x dx\). Therefore, the integral becomes \(\int 2 u du\), which evaluates to \(u^2 + C\). Substituting \(u = \sin x\) back in, we get \(\sin^2 x + C\).

Solving integral with the substitution \(u=\cos x\)

Now consider the substitution \(u=\cos x\). This gives \(du = -\sin x dx\). The integral becomes \(-\int 2 u du\), which evaluates to \(-u^2 + C\). Substituting \(u=\cos x\) back in, we get \(-\cos^2 x + C\).

Explanation of equivalence

At first glance, these two solutions \(\sin^2 x + C\) and \(-\cos^2 x + C\) may seem different. However, from the Pythagorean Identity in trigonometry, we know \(\sin^2 x + \cos^2 x = 1\). Therefore, either of these expressions can be written in terms of the other. More specifically, \(\sin^2 x = 1- \cos^2x\) and \(-\cos^2x= - (1-\sin^2x)\) or \(\sin^2x - 1\). Hence, the results are indeed equivalent.

Key concepts

Substitution Method in Integrals

The substitution method is a powerful tool in calculus used to simplify integrals by changing the variable of integration. It's also referred to as 'u-substitution'. To implement this method effectively, choose a part of the integral to represent with a new variable, usually 'u', that makes the integral easier to handle.

For instance, when given an integral involving trigonometric functions like \( \int 2 \sin x \cos x \, dx \), we can set \( u = \sin x \) or \( u = \cos x \). The differential \( du \) is then the derivative of the chosen 'u' with respect to 'x', multiplied by \( dx \). This transforms the integral into a new form in terms of 'u' that is usually easier to integrate.

Example of 'u' substitution

If we choose \( u = \sin x \), then \( du = \cos x \, dx \), and the integral becomes \( \int 2u \, du \), a simpler integral to evaluate. Upon solving and reverting back to the original variable 'x', we obtain the integrated function in terms of the original variable.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved angles. They are indispensable tools for simplifying and evaluating expressions and integrals involving trigonometric functions.

Common identities include the Pythagorean Identity, \( \sin^2 x + \cos^2 x = 1 \), and angle sum and difference formulas. In the context of integrating functions like \( 2 \sin x \cos x \), these identities allow us to express trigonometric functions in alternative forms which can be more conducive to integration.

Applying an Identity in Integration

For example, using the Pythagorean Identity, one can convert \( \sin^2 x \) into \( 1 - \cos^2 x \) and vice versa. This is crucial when verifying that two different-looking integral results are, in fact, equivalent, as they are merely different expressions of the same trigonometric relationship.

Evaluating Definite Integrals

Evaluating definite integrals is the process of finding the exact value of an integral over a specific interval. Unlike indefinite integrals, which include a constant of integration \( C \), definite integrals result in a numerical value that represents the net area between the curve of the function and the x-axis, between two points.

In many cases, the evaluation involves finding the antiderivative of the function and applying the Fundamental Theorem of Calculus, which entails substituting the upper and lower bounds of integration into the antiderivative and taking the difference.

Definite Integral Example

Suppose you're asked to evaluate the definite integral \( \int_a^b 2 \sin x \cos x \, dx \). Using substitution, we would find the indefinite integral and then calculate the difference between the values of the antiderivative at \( b \) and \( a \). Such evaluation is essential for understanding the behavior of functions and the geometry of their graphs over an interval.