Problem 307 - Calculus Volume 1

Question

In the following exercises, use a change of variables to show that each definite integral is equal to zero. $$ \int_{0}^{\sqrt{\pi}} t \cos \left(t^{2}\right) \sin \left(t^{2}\right) d t $$

Step-by-Step Solution

Answer

The integral evaluates to 0 by using a substitution \( u = t^2 \).

Step 1: Identify the Change of Variables

We begin by setting a substitution to simplify the integration. Let \( u = t^2 \), then the differential \( du = 2t \, dt \), or \( dt = \frac{du}{2t} \). Since \( t \) can be written as \( \sqrt{u} \), we substitute \( t \) in terms of \( u \) in the integral.

Step 2: Change the Limits of Integration

With the substitution \( u = t^2 \):- When \( t = 0 \), \( u = 0^2 = 0 \).- When \( t = \sqrt{\pi} \), \( u = (\sqrt{\pi})^2 = \pi \).Thus, the new limits for \( u \) are from 0 to \( \pi \).

Step 3: Substitute the Variables in the Integral

Substitute \( t \) and \( dt \) in the integral:\[ \int_{0}^{\pi} \sqrt{u} \cdot \cos(u) \cdot \sin(u) \cdot \frac{du}{2\sqrt{u}} \]This simplifies to:\[ \frac{1}{2} \int_{0}^{\pi} \cos(u) \sin(u) \, du \]

Step 4: Simplify the Integral and Apply Trigonometric Identity

Recognize the trigonometric identity \( \cos(u)\sin(u) = \frac{1}{2}\sin(2u) \). Apply this identity:\[ \frac{1}{2} \int_{0}^{\pi} \cos(u) \sin(u) \, du = \frac{1}{2} \cdot \frac{1}{2} \int_{0}^{\pi} \sin(2u) \, du \]This simplifies to:\[ \frac{1}{4} \int_{0}^{\pi} \sin(2u) \, du \]

Step 5: Evaluate the Integral

The integral of \( \sin(2u) \) is \( -\frac{1}{2} \cos(2u) \). Evaluating from 0 to \( \pi \) gives:\[ \frac{1}{4} \left[ -\frac{1}{2} \cos(2u) \right]_{0}^{\pi} = \frac{1}{4} \left[ -\frac{1}{2} \cos(2\pi) + \frac{1}{2} \cos(0) \right] = \frac{1}{4} \cdot 0 = 0 \]

Step 6: Conclusion

The integral evaluates to 0. Therefore, the definite integral \( \int_{0}^{\sqrt{\pi}} t \cos(t^2) \sin(t^2) \, dt = 0 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial tools in calculus, especially when solving integrals involving trigonometric functions. They allow you to simplify complex expressions into more manageable forms. In our example integral, we used the identity:

• \( \cos(u)\sin(u) = \frac{1}{2}\sin(2u) \)

This identity states that the product of sine and cosine of the same angle can be rewritten as half the sine of the double angle. It's a powerful way to reduce the complexity of the integral. By using this identity, the original expression \( \cos(u)\sin(u) \) readily transforms, making it easier to evaluate. Such identities are standard within trigonometry and calculus, offering a streamlined approach to solving problems. They save time and reduce errors, making them indispensable in both mathematical theory and practice.

Change of Variables

The change of variables, also known as u-substitution, is a technique used to simplify integrals by turning a complex function into a simpler one. In our solution, we use this method by substituting \( u = t^2 \). This transforms the original integral into one that's easier to work with.

• First, the differential changes accordingly: starting with \( du = 2t \, dt \), we rearrange to find \( dt = \frac{du}{2t} \).
• The variable \( t \) itself is changed to \( \sqrt{u} \), linking it directly to our substitution variable \( u \).

This process allows us to shift the problem into a different, often simpler form. Moreover, changing variables can help in aligning the limits of integration with the new function introduced by the substitution. This method is a staple in integration techniques, as it often turns unsolvable integrals into those with straightforward solutions.

Limits of Integration

When performing integration, adjusting the limits is necessary whenever the variable changes. Using the substitution \( u = t^2 \), the original variable \( t \) has limits from 0 to \( \sqrt{\pi} \). For the variable \( u \), the limits transform as follows:

• When \( t = 0 \), since \( u = t^2 \), we find \( u = 0^2 = 0 \).
• When \( t = \sqrt{\pi} \), using the same logic, \( u = (\sqrt{\pi})^2 = \pi \).

Hence, the limits for \( u \) in the integral shift from 0 to \( \pi \). Adjusting the limits ensures the integration bounds correspond to the current variable, maintaining consistency in evaluations. It is a critical step since incorrect limits can lead to erroneous results. Always remember to update limits timely and accurately in response to substitutions, as they are fundamental to evaluating definite integrals correctly.