Textbook Question
Evaluating integrals Evaluate the following integrals.
∫ d𝓍/[(tan⁻¹ 𝓍) (1 + 𝓍²)]
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7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
1:59 minutesProblem 8.1.1
Textbook Question
What change of variables would you use for the integral ∫(4 - 7x)^(-6) dx?
Verified step by step guidance1
Step 1: Recognize that the integral involves a composite function (4 - 7x)^(-6). To simplify the integration, use a substitution method.
Step 2: Let u = 4 - 7x. This substitution simplifies the expression inside the integral. Compute the derivative of u with respect to x: du/dx = -7.
Step 3: Rearrange the derivative to express dx in terms of du: dx = du / (-7).
Step 4: Substitute u and dx into the integral. The integral becomes ∫u^(-6) * (du / -7).
Step 5: Factor out the constant -1/7 from the integral, leaving ∫u^(-6) du. This integral can now be solved using the power rule for integration.
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Change of Variables
Change of variables, also known as substitution, is a technique used in integration to simplify the integral by transforming it into a more manageable form. This involves substituting a new variable for an expression in the integral, which can make the integration process easier. The goal is to find a relationship between the original variable and the new variable that simplifies the integrand.
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Integration Techniques
Integration techniques are methods used to evaluate integrals that may not be solvable using basic antiderivatives. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these techniques is essential for tackling complex integrals, as they provide strategies to break down the problem into simpler parts.
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Power Rule for Integration
The power rule for integration states that the integral of x raised to the power n (where n ≠ -1) is (x^(n+1))/(n+1) + C, where C is the constant of integration. This rule is fundamental in calculus and is often applied after performing a change of variables to simplify the integrand into a polynomial form, making it easier to integrate.
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