Problem 29 Evaluate the integral. $$ \i... [FREE SOLUTION]

Chapter 5: Problem 29

Evaluate the integral. $$ \int_{\pi / 2}^{\pi} \frac{\sin x}{1+\cos ^{2} x} d x $$

Short Answer

Expert verified

The value of the integral is $\pi/4$.

Step by step solution

Set up the substitution

Start with selecting a substitution that simplifies the equation. Here, choose $u = \cos x$. Thus, $du=-\sin x\, dx$.

Change the bounds

Perform the calculation of the new bounds after substitution using $u = \cos x$. When $x=\pi/2$, $u= \cos(\pi/2) = 0$ and when $x= \pi$, $u= \cos(\pi) = -1$. So the bounds change from $\pi/2$ and $\pi$ to 0 and -1 respectively.

Substitute and simplify

Next, replace $dx$ and $\sin x$ in terms of $du$ and $u$. So, the integral becomes $-\int_{0}^{-1} \frac{1}{1+u^2} du$. The bounds are from 0 to -1 which can be reversed to become from -1 to 0 and hence the sign becomes positive.

Evaluate the integral

Now evaluate $\int_{-1}^{0} \frac{1}{1+u^2} du$. As we know, $\int \frac{1}{1+u^2} du = \arctan(u) + C$, thus after applying the limits, this gives $\arctan(0) - \arctan(-1) = 0 - (-\pi/4) = \pi/4$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Substitution

Trigonometric substitution is a clever technique used to simplify the integration of expressions involving square roots or trigonometric functions. In this exercise, we selected the substitution $ u = \cos x $. This choice was beneficial because the derivative $ du = -\sin x \, dx $ replaced $ \sin x \, dx $ seamlessly. The goal of trigonometric substitution is to transform the integral into a simpler form to work with.

Look for patterns involving trigonometric identities.
Select a substitution that simplifies the integral.
Replace all occurrences of the original variable with the new one involving the trigonometric function.

Trigonometric substitution often leads to simpler bounds and integrals that are more manageable to evaluate.

Change of Bounds

Changing the bounds is a crucial part of integration by substitution. Since the integration variable changes, the limits of integration must also be appropriately updated. When we substituted $ u = \cos x $ in the original integral, the bounds had to be changed as follows:

Original lower bound: $ x = \frac{\pi}{2} $ results in $ u = \cos \left(\frac{\pi}{2}\right) = 0 $.
Original upper bound: $ x = \pi $ gives $ u = \cos(\pi) = -1 $.

This process helps in maintaining the integrity of the integral during the transformation. Always carefully compute the new bounds to avoid errors in the integral evaluation.

Integration Techniques

Various integration techniques come to play during definite integral evaluations. In this problem, after performing trigonometric substitution, simplifying the integral was key. We arrived at:
\[ \int_{0}^{-1} \frac{1}{1+u^2} \, du \] Simplification turned it into a form that was straightforward to integrate using standard techniques. These include:

Reversing the bounds when necessary 鈥� in this case, reversing from $[-1, 0]$ to $[0, -1]$ made the integral positive.
Recognizing the formula applicable, $ \int \frac{1}{1+u^2} \, du = \arctan(u) + C $, which is a direct result of inverse trigonometric integration.

Mastery of these techniques facilitates quicker and more accurate integration.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in calculus, especially in integration. For this integral, the expression $ \int \frac{1}{1+u^2} \, du = \arctan(u) + C $ was used. Using an inverse trigonometric function simplifies the evaluation, particularly in this form:

The antiderivative of $ \frac{1}{1+u^2} $ is the arc tangent function, $ \arctan(u) $.
This allows the integral to be evaluated directly, especially with defined limits.

Once we applied the limits, we found $ \arctan(0) - \arctan(-1) = \pi / 4 $. Understanding inverse trigonometric functions is crucial for easily solving integrals that are otherwise complex.

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91影视

Evaluate the integral. $$ \int_{\pi / 2}^{\pi} \frac{\sin x}{1+\cos ^{2} x} d x $$

Short Answer

Step by step solution

Set up the substitution

Change the bounds

Substitute and simplify

Evaluate the integral

Key Concepts

Trigonometric Substitution

Change of Bounds

Integration Techniques

Inverse Trigonometric Functions

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