Problem 107 - Calculus for Biology and Medicine

Question

Evaluate the definite integrals. $$ \int_{0}^{\pi / 4} \sin \left(x-\frac{\pi}{4}\right) d x $$

Step-by-Step Solution

Answer

The integral evaluates to 0.

Step 1: Rewrite the Integrand Using Sine Angle Addition Formula

The integrand is given as \( \sin \left(x-\frac{\pi}{4}\right) \). We can use the sine angle addition formula, which states \( \sin(a-b) = \sin a \cos b - \cos a \sin b \). Substitute \( a = x \) and \( b = \frac{\pi}{4} \):\[ \sin \left(x-\frac{\pi}{4}\right) = \sin x \cos \frac{\pi}{4} - \cos x \sin \frac{\pi}{4} \]Given \( \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \), the expression becomes:\[ \sin x \cdot \frac{\sqrt{2}}{2} - \cos x \cdot \frac{\sqrt{2}}{2} \]

Step 2: Separate the Integral

Now that we have rewritten the integrand, separate the integral into two distinct parts:\[ \int_{0}^{\pi / 4} \left( \frac{\sqrt{2}}{2} \sin x - \frac{\sqrt{2}}{2} \cos x \right) dx = \frac{\sqrt{2}}{2} \int_{0}^{\pi/4} \sin x \, dx - \frac{\sqrt{2}}{2} \int_{0}^{\pi/4} \cos x \, dx \]

Step 3: Evaluate the Integral of \(\sin x\)

Calculate the integral of \( \sin x \):\[ \int \sin x \, dx = -\cos x \]Evaluate this from 0 to \( \pi/4 \):\[ -\cos \left(\frac{\pi}{4}\right) + \cos(0) = -\frac{\sqrt{2}}{2} + 1 \]

Step 4: Evaluate the Integral of \(\cos x\)

Calculate the integral of \( \cos x \):\[ \int \cos x \, dx = \sin x \]Evaluate this from 0 to \( \pi/4 \):\[ \sin \left(\frac{\pi}{4}\right) - \sin(0) = \frac{\sqrt{2}}{2} \]

Step 5: Substitute and Combine Results

Substitute the values obtained in Steps 3 and 4 back and apply the constants:For \( \int_{0}^{\pi/4} \sin x \, dx \) result:\[ \frac{\sqrt{2}}{2} \left(-\frac{\sqrt{2}}{2} + 1 \right) \]For \( \int_{0}^{\pi/4} \cos x \, dx \) result:\[ \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{1}{2} \]Now, combine the results:\[ \frac{\sqrt{2}}{2} \left(-\frac{\sqrt{2}}{2} + 1 \right) - \frac{1}{2} \]Calculate:\[ \frac{\sqrt{2}}{2} \cdot 1 - \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \]\[ = \frac{\sqrt{2}}{2} - \frac{1}{2} - \frac{1}{2} = 0 \]

Step 6: Final Calculation and Conclusion

After simplifying, the final result of the definite integral is:\[ \int_{0}^{\pi / 4} \sin \left(x-\frac{\pi}{4}\right) dx = 0 \]

Key Concepts

Trigonometric Integrals

Trigonometric integrals are a specific type of integral where the function being integrated is related to trigonometric functions, such as sine, cosine, etc. These types of integrals often appear in calculus, and solving them is essential for understanding the behavior of oscillatory functions.
They are especially important in advanced mathematics and fields like physics and engineering.

When faced with a trigonometric integral, like the one in our exercise, it's important to consider simplification techniques such as trigonometric identities. These identities can transform complex expressions into more manageable ones.

• Identifying patterns: Before integration, look for patterns or identities that could simplify the integral.
• Breaking down: Trigonometric identities often allow you to decompose a function into simpler parts that are easy to integrate separately.

This process helps not only in simplifying calculations but also in visualizing the integral as a combination of simpler functions.

Sine and Cosine Functions

The sine and cosine functions are fundamental in trigonometry and appear very frequently in integration problems. They describe the relationship between the angles in a right-angle triangle and the ratios of its sides.

In terms of the unit circle,

• Sine function: Given by \( ext{sin}(x) \), it represents the y-coordinate of a point on the unit circle.
• Cosine function: Given by \( ext{cos}(x) \), it represents the x-coordinate.

The interval from 0 to \( \frac{\pi}{2} \) on the circle is especially key for understanding these functions.
In our exercise:

• Understanding how each function behaves over the interval allows us to compute the integral effectively.
• Examining patterns in these functions aids in predicting values of integrals without direct computation, which can be handy in exams and real-life applications.

Angle Addition Formulas

The angle addition formulas are invaluable tools in calculus for breaking down complex trigonometric expressions into simpler parts. The formula for sine, \( \sin(a - b) = \sin a \cos b - \cos a \sin b \, \) helps us rewrite expressions involving sine and cosine of differences of angles.

Let's explore why angle addition formulas are important:

• Simplification: They allow the conversion of complex expressions into forms that are easier to integrate, as demonstrated in the exercise where \( \sin(x - \frac{\pi}{4}) \) becomes \( \sin x \cdot \frac{\sqrt{2}}{2} - \cos x \cdot \frac{\sqrt{2}}{2} \).
• Versatility: These formulas are applicable in numerous scenarios such as solving inequalities, trigonometric equations, and evaluating derivatives.

This method of rewriting makes it easier to handle definite integrals, facilitating separate integration components leading to a successful evaluation of even seemingly tricky integrals.