Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 8: Problem 35
Use a substitution to change the integral into one you can find in the table. Then evaluate the integral. \(\int \frac{d y}{y \sqrt{3+(\ln y)^{2}}}\)
Short Answer
Expert verified
\( \ln |\ln y + \sqrt{3 + (\ln y)^2}| + C \)
Step by step solution
01
Choose a Substitution
To simplify the integral, let's choose a substitution that simplifies the expression under the square root. Set \( u = \ln y \). This substitution is appropriate because the expression inside the square root is \( 3 + (\ln y)^2 \).
02
Differentiate and Convert Integral
Differentiate the substitution with respect to \( y \): \( du = \frac{1}{y} \, dy \). This implies \( dy = y \, du \). Substitute these into the integral, yielding:\[ \int \frac{dy}{y \sqrt{3 + (\ln y)^2}} = \int \frac{y \, du}{y \sqrt{3 + u^2}} = \int \frac{du}{\sqrt{3 + u^2}} \].
03
Recognize The Integral Form
The integral \( \int \frac{du}{\sqrt{3 + u^2}} \) is a standard integral that can be found in integral tables. It is equivalent to the form \( \int \frac{dx}{\sqrt{a^2 + x^2}} = \ln |x + \sqrt{a^2 + x^2}| + C \), where \( a^2 = 3 \).
04
Evaluate the Integral
Using the formula from the table, evaluate the integral:\[ \int \frac{du}{\sqrt{3 + u^2}} = \ln |u + \sqrt{3 + u^2}| + C \].
05
Replace the Substitution Variable
Return to the original variable by replacing \( u \) with \( \ln y \):\[ \ln |\ln y + \sqrt{3 + (\ln y)^2}| + C \].
06
Write the Final Result
Hence, the evaluated integral is:\[ \ln |\ln y + \sqrt{3 + (\ln y)^2}| + C \].
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Integrals
An integral is a fundamental concept in calculus that allows us to determine the area under a curve or the total accumulation of change, among other applications. When you're asked to evaluate an integral, you are finding a function whose derivative gives you the original function under the integral sign. This is sometimes referred to as finding an "antiderivative."
Integrals can be definite or indefinite:
Integrals can be definite or indefinite:
- Definite Integrals: These have upper and lower limits, representing the specific area under a curve within those boundaries.
- Indefinite Integrals: These lack limits and denote a general form of antiderivative with a constant of integration, denoted as 'C'.
Integration by Substitution
Integration by substitution is a powerful technique used to simplify complex integrals by transforming them into simpler forms. The method is akin to the process of reversing the chain rule from differentiation.
Here's a quick breakdown of how it works:
Here's a quick breakdown of how it works:
- Choose a Substitution (u): Identify a part of the integrand that can be replaced with a simpler variable, usually denoted as 'u'. This is often chosen to simplify a complex expression or an inner function.
- Differentiate u: Once you have your substitution, differentiate it with respect to the original variable to find 'du'. This will help in rewriting the integral.
- Replace and Simplify: Substitute the expressions in the original integral with 'u' and 'du', which should simplify the integral. Solve the simplified integral.
- Back-substitute: Finally, convert back to the original variable to express the solution in terms of the variable given in the problem.
Utilizing Integral Tables
Integral tables are a handy reference tool in calculus that list common integrals and their antiderivatives. These tables save a significant amount of time by providing solutions to integrals with recognized forms.
Here's how you can effectively use these tables:
Here's how you can effectively use these tables:
- Recognize the Form: Identify the structure of your integral and try to match it with one of the forms listed in the integral table. For example, the integral form \( \int \frac{dx}{\sqrt{a^2 + x^2}} \) is found in many standard tables.
- Make Necessary Adjustments: You may need to manipulate the integral through methods like substitution to fit the standard form exactly. This often involves adjusting constants or re-writing terms within the integral.
- Apply the Formula: Once the integral matches a form from the table, apply the corresponding formula to solve it effortlessly, as illustrated when we found the antiderivative \( \ln |u + \sqrt{3 + u^2}| + C \) using a known integral result.
