Problem 52 - Calculus for Biology and Medicine

Question

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{0}^{\pi / 4} \frac{\sin x}{\cos ^{2} x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \sqrt{2} - 1 \).

Step 1: Identify a Suitable Substitution

To solve the integral \( \int_{0}^{\pi/4} \frac{\sin x}{\cos^2 x} dx \) using substitution, we'll start by identifying a substitution that simplifies the integrand. Notice that the derivative of \( \cos x \) is \( -\sin x \), so we can use the substitution \( u = \cos x \), which leads to \( du = -\sin x \, dx \). This transforms our integral into one involving \( u \).

Step 2: Change the Limits of Integration

With the substitution \( u = \cos x \), we need to change the limits of integration from \( x \) values to \( u \) values. When \( x = 0 \), \( u = \cos 0 = 1 \); and when \( x = \pi/4 \), \( u = \cos(\pi/4) = \frac{1}{\sqrt{2}} \). The new limits for \( u \) are from 1 to \( \frac{1}{\sqrt{2}} \).

Step 3: Substitute and Simplify the Integral

Substitute the original integrand with respect to \( u \). The integrand \( \frac{\sin x}{\cos^2 x} dx \) becomes \( -\frac{1}{u^2} du \) using \( du = -\sin x \, dx \) and \( \cos x = u \). The integral now looks like \(-\int_{1}^{1/\sqrt{2}} \frac{1}{u^2} \, du \).

Step 4: Evaluate the Integral

The integral \(-\int \frac{1}{u^2} \, du \) can be rewritten as \(-\int u^{-2} \, du \). Integrating gives \( -(-u^{-1}) = u^{-1} \). Evaluate from 1 to \( 1/\sqrt{2} \), resulting in \[ \left[ \frac{1}{u} \right]_{1}^{1/\sqrt{2}} = \left( \sqrt{2} - 1 \right) \].

Step 5: Simplify the Result

Compute \( \sqrt{2} - 1 \) to get the final numerical value of the definite integral. The solution is \( \sqrt{2} - 1 \).

Key Concepts

Substitution Method

The substitution method in integration is like finding a clever way to simplify an integral, making it easier to solve. When tackling trigonometric integrals, a useful strategy is to look for a function within the integrand whose derivative is also present. For instance, in the integral \( \int \frac{\sin x}{\cos^2 x} \, dx \), we notice that the derivative of \( \cos x \) is \( -\sin x \). This observation guides us to make a substitution: \( u = \cos x \). By doing so, we transform the integral into the variable \( u \), taking advantage of the derivative relation.Substitution involves a few key steps:

• Identify the substitution: Find a part of the integrand whose derivative makes the problem simpler.
• Express \( dx \) in terms of \( du \): Use the derivative to write \( dx = \frac{du}{-\sin x} \).
• Transform the integrand: Replace terms to express the entire integrand in terms of \( u \) and \( du \).
• Change the limits of integration: Substitute the original \( x \) limits with corresponding \( u \) values.

Using substitution can turn a complex problem into a much simpler one, making it a vital tool in solving integrals.

Integral Calculus

Integral calculus is the mathematical study focused on finding the total accumulation of quantities, typically areas under curves. In the context of definite integrals, you not only find an antiderivative, but also evaluate it over a specific interval. This gives a precise value representing, for instance, the total area between a function and the x-axis within given bounds. To solve definite integrals, follow these steps:

• Determine the appropriate method, like substitution or integration by parts.
• Find the antiderivative: This often involves algebraic manipulation or recognizing basic integral forms.
• Apply the limits of integration: Substitute the upper and lower limits into the antiderivative, and find the difference.

Definite integral calculations offer more than just solutions to functions. They help us understand critical real-world applications, such as calculating distances and areas, which are fundamental in physics and engineering.

Trigonometric Integrals

Trigonometric integrals are those involving trigonometric functions like \( \sin(x) \), \( \cos(x) \), and others. Solving these integrals often requires clever substitutions or identities to simplify the integrand. In our example, \( \int_{0}^{\pi/4} \frac{\sin x}{\cos^2 x} \, dx \), the integral naturally lends itself to substitution because the derivative of the trigonometric functions are related.When approaching trigonometric integrals, consider these tips:

• Look for direct derivatives: Trigonometric functions often have derivatives that appear elsewhere in the integrand, suggesting a simple substitution.
• Utilize identities: Know basic trigonometric identities, which can help transform the integrand into a more manageable form.
• Consider symmetry: Some trigonometric integrals can be simplified by recognizing symmetrical properties.

By mastering these techniques, you can solve complex trigonometric integrals more easily. Understanding how to manipulate these functions is essential in many fields of science and engineering.