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Chapter 7: Problem 26

evaluate the integral, and check your answer by differentiating. $$\int \frac{d y}{\csc y}$$

Short Answer

Expert verified
The integral is \(-\cos y + C\).

Step by step solution

01

Simplify the Integral

The given integral is \( \int \frac{d y}{\csc y} \). Recalling that \( \csc y = \frac{1}{\sin y} \), we rewrite the integral as \( \int \sin y \, d y \).
02

Integrate

Now, we need to find the integral of \( \sin y \). The integral of \( \sin y \) is \( -\cos y + C \), where \( C \) is the constant of integration. Therefore, \( \int \sin y \, d y = -\cos y + C \).
03

Differentiate to Check the Answer

To verify our answer, differentiate \( -\cos y + C \) with respect to \( y \). The derivative of \( -\cos y \) is \( \sin y \), and the derivative of any constant \( C \) is 0. Therefore, the derivative of \( -\cos y + C \) is \( \sin y \), which matches the integrand, confirming the result.

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Key Concepts

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

Integration Techniques
Integration is a core idea in calculus and involves finding the integral or antiderivative of a function. When approaching integrals, there are several techniques you can use to simplify the process. One basic technique is substitution, wherein you make a clever substitution to simplify the integral to a form that is easier to handle. In our exercise, we used a trigonometric simplification. Given the integral \( \int \frac{d y}{\csc y} \), we leveraged the identity \( \csc y = \frac{1}{\sin y} \) to rewrite it as \( \int \sin y \, d y \).
  • Substitution: help transform the integral into a simpler form.
  • Rewriting using identities: Use trigonometric identities to simplify the expression you are integrating.

Understanding different techniques can greatly aid in evaluating integrals efficiently and correctly.
Trigonometric Identities
Trigonometric identities are expressions that relate the angles and sides of triangles. These identities are handy in simplifying trigonometric expressions.
In the given problem, we used the cosecant identity \( \csc y = \frac{1}{\sin y} \) to rewrite the original expression. This process facilitated easier integration. There are many identities that can be crucial, like:
  • Pythagorean Identities: such as \( \sin^2 y + \cos^2 y = 1 \)
  • Reciprocal Identities: like \( \csc y = \frac{1}{\sin y} \)
  • Quotient Identities: such as \( \tan y = \frac{\sin y}{\cos y} \)

Giving attention to these identities helps simplify the integration process and ensures that the solution is both correct and clear.
Differentiation
Differentiation is the process of finding the derivative of a function, revealing how it changes. It鈥檚 the reverse process of integration. In this exercise, we differentiate our result to confirm its correctness.
After integrating \( \int \sin y \, dy = -\cos y + C \), we differentiated \( -\cos y + C \) to get \( \sin y \). This confirms the initial function is indeed the correct antiderivative.
  • The derivative of \( \sin y \) is \( \cos y \), and the derivative of \( \cos y \) is \( -\sin y \).
  • Derivatives of constants are zero because constants do not change.

Using differentiation to check the results is a robust technique to verify if the integration was performed correctly.

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Most popular questions from this chapter

By writing out the sums, determine whether the following are valid identities. A. $$\int\left[\sum_{i=1}^{n} f_{i}(x)\right] d x=\sum_{i=1}^{n}\left[\int f_{i}(x) d x\right]\( B. $$\frac{d}{d x}\left[\sum_{i=1}^{n} f_{i}(x)\right]=\sum_{i=1}^{n}\left[\frac{d}{d x}\left[f_{i}(x)\right]\right]\)

When part of each term of a sum cancels part of the next term, leaving only portions of the first and last terms at the end. the sum is said to telescope. In Exercises \(43-46,\) evaluate the telescoping sum. $$\sum_{k=1}^{30}\left(\frac{1}{k}-\frac{1}{k+1}\right)$$

(a) Make a conjecture about the value of the limit $$\lim _{k \rightarrow 0} \int_{1}^{x} t^{k-1} d t \quad(x>0)$$ (b) Check your conjecture by evaluating the integral, and then using L'H么pital's rule to find the limit.

When part of each term of a sum cancels part of the next term, leaving only portions of the first and last terms at the end. the sum is said to telescope. In Exercises \(43-46,\) evaluate the telescoping sum. $$\sum_{k=1}^{100}\left(2^{k+1}-2^{k}\right)$$

Let \(F(x)=\int_{0}^{x} \frac{t-3}{t^{2}+7} d t\) for \(-\infty
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