Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals. $$\int_{\pi / 4}^{\pi / 2} \frac{\cos x}{\sin ^{2} x} d x$$

Short answer

Question: Evaluate the definite integral of \(\frac{\cos x}{\sin^2 x}\) with respect to \(x\) over the interval \([\frac{\pi}{4}, \frac{\pi}{2}]\). Answer: The definite integral is equal to \(-1 + \sqrt{2}\).

Step-by-step solution

Introduce the substitution

Let's introduce the substitution \(u = \sin x\). Then, the derivative of \(u\) with respect to \(x\) is \(\frac{du}{dx} = \cos x\). So, we can write \(\cos x \, dx = du\). Our integral becomes: $$\int \frac{1}{u^2} du$$

Integrate with respect to u

Now, let's find the indefinite integral of \(\frac{1}{u^2}\) with respect to \(u\): $$\int \frac{1}{u^2} du = -\frac{1}{u} + C$$ (where C is the constant of integration).

Replace u with original variable

Now, substitute back \(u = \sin x\) to get the integral in terms of \(x\): $$-\frac{1}{\sin x} + C$$

Use the Fundamental Theorem of Calculus to evaluate the definite integral

Now that we have the antiderivative, we can use the Fundamental Theorem of Calculus to compute the definite integral from \(\frac{\pi}{4}\) to \(\frac{\pi}{2}\): $$\int_{\pi / 4}^{\pi / 2} \frac{\cos x}{\sin ^{2} x} d x=-\frac{1}{\sin x} \Big|^{\frac{\pi}{2}}_{\frac{\pi}{4}} = \left(-\frac{1}{\sin (\frac{ \pi}{2})}\right) - \left(-\frac{1}{\sin (\frac{\pi}{4})}\right)$$

Compute the definite integral

Finally, let's compute the integral: $$\left(-\frac{1}{\sin (\frac{ \pi}{2})}\right) - \left(-\frac{1}{\sin (\frac{\pi}{4})}\right) = \left(-\frac{1}{1}\right) - \left(-\frac{1}{\frac{\sqrt{2}}{2}}\right) = -1 + \sqrt{2}$$ Therefore, the definite integral is equal to: $$\int_{\pi / 4}^{\pi / 2} \frac{\cos x}{\sin ^{2} x} d x= -1 + \sqrt{2}$$

Key concepts

Change of Variables

The change of variables, also known as substitution, is a powerful technique used when working with integrals. It helps simplify the integration process by transforming a challenging integral into a more manageable form. Imagine you're given an integral that's tough to crack. By changing variables, you replace parts of the integrand (the function inside the integral) with a substitution that makes the math more straightforward.

In practice, this often involves substituting a function of one variable for another, such as replacing a trigonometric expression with a single variable. This was done in the exercise by substituting \( u = \sin x \), changing the integral to a simpler form based on \( u \). This replacement leads to a different integral, \( \int \frac{1}{u^2} \, du \), which is much simpler to integrate.

In essence, changing variables is about finding a new perspective. This new view can often make the problem more approachable, especially when original expressions are complex. Remember, once you've evaluated the integral with the new variable, you must substitute back to the original variable to complete the process.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration, two cornerstone topics in calculus. It provides a way to evaluate definite integrals, which are used to calculate the accumulated quantity, like area under a curve.

There are two parts: the first part deals with the relationship between differentiation and integration, while the second part is what we use to compute definite integrals. The second part states that if a function is continuous on a closed interval \([a, b]\) and \( F \) is an antiderivative of function \( f \), then the integral of \( f \) from \( a \) to \( b \) is given by the difference \( F(b) - F(a) \).

In the exercise, this theorem allows us to evaluate the original definite integral after finding the antiderivative \(-\frac{1}{\sin x}\). We simply calculate values at the limits \(\pi/2\) and \(\pi/4\), then determine the difference to find the exact area under the curve for those bounds. By employing this theorem, we translate the process of integration into an algebraic computation, making it very practical and efficient.

Integration by Substitution

Integration by substitution is closely linked to the change of variables technique and is extremely helpful for definite and indefinite integrals. It allows us to turn a difficult-looking integral into a simpler one by making an appropriate substitution.

Here's a step-by-step look into how it works:

• First, identify a part of the integrand that can be substituted, setting this equal to a new variable, such as \( u \).
• Calculate the derivative, or differential, of this substitution, \( du \).
• Transform the integral into terms of the new variable \( u \) and \( du \). This often simplifies the integrand significantly.
• Find the antiderivative in terms of \( u \).
• Finally, substitute back the original variable to obtain the solution in familiar terms.

In this exercise, starting with \( u = \sin x \) transforms the problem from working with trigonometric functions to dealing with a rational function of \( u \), immensely simplifying the integration. This substitution is the key to unlocking a simpler integral, which can then be solved more comfortably. With definite integrals, remember to update the bounds to match the substitution, or revert to the original variable before evaluating.