Textbook Question
Multiple substitutions If necessary, use two or more substitutions to find the following integrals.
β«β^Ο/Β² (cos ΞΈ sin ΞΈ) / β(cosΒ² ΞΈ + 16) dΞΈ (Hint: Begin with u = cos ΞΈ .)
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8. Definite Integrals
Substitution
1:56 minutesProblem 5.5.5
Textbook Question
When using a change of variables u = g(π) to evaluate the definite integral β«βα΅ Ζ(g(π)) g' (π) d(π), how are the limits of integration transformed?
Verified step by step guidance1
Recognize that when performing a substitution in a definite integral, the variable of integration changes from \( x \) to \( u = g(x) \).
The original integral is \( \int_a^b f(g(x)) g'(x) \, dx \). After substitution, \( dx \) is replaced by \( \frac{du}{g'(x)} \), but since \( du = g'(x) dx \), the integral becomes \( \int_{u(a)}^{u(b)} f(u) \, du \).
To find the new limits of integration, evaluate the substitution function \( g(x) \) at the original limits: the lower limit \( a \) transforms to \( u(a) = g(a) \), and the upper limit \( b \) transforms to \( u(b) = g(b) \).
Thus, the definite integral with respect to \( x \) from \( a \) to \( b \) is equivalent to the integral with respect to \( u \) from \( g(a) \) to \( g(b) \).
This change of limits ensures the integral remains consistent under the substitution and allows you to evaluate the integral in terms of \( u \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Change of Variables (Substitution) in Integration
This technique simplifies integrals by substituting a new variable u = g(x), transforming the integral into terms of u. It helps to rewrite complex integrals into more manageable forms by changing the variable of integration.
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Derivative of the Substitution Function
When substituting u = g(x), the differential dx is replaced by du = g'(x) dx. This derivative g'(x) adjusts the integrand to maintain equivalence between the original and transformed integrals.
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Transformation of Limits of Integration
In definite integrals, the original limits a and b in terms of x must be converted to new limits in terms of u by evaluating u = g(a) and u = g(b). This ensures the integral's bounds correspond correctly to the substituted variable.
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