Math 152 metric IHU 3. In this problem we'll evaluate the integral cos²(z) sin³ (2) dr. (a) First, we'll re-write sin³ (2) as sin²(x) sin(r). Write the integral with this change made. (b) Now, we'll convert sin²(z) to an expression that involves cosine instead. Use the identity sin² 0 + cos² 0 = 1 to replace sin2 (r) with something involving cosines, and w the integral with the change made. (c) At this point we can use a u-substitution to make this integral much easier. Substitute U= cos(x). Write the new integral completely in terms of u. (d) Distribute any multiplication inside your integral, then integrate and complete the integral.
Math 152 metric IHU 3. In this problem we'll evaluate the integral cos²(z) sin³ (2) dr. (a) First, we'll re-write sin³ (2) as sin²(x) sin(r). Write the integral with this change made. (b) Now, we'll convert sin²(z) to an expression that involves cosine instead. Use the identity sin² 0 + cos² 0 = 1 to replace sin2 (r) with something involving cosines, and w the integral with the change made. (c) At this point we can use a u-substitution to make this integral much easier. Substitute U= cos(x). Write the new integral completely in terms of u. (d) Distribute any multiplication inside your integral, then integrate and complete the integral.
Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:**Trigonometric Integrals**
In this problem, we'll evaluate the integral \(\int \cos^2(x) \sin^3(x) \, dx\).
(a) First, we’ll re-write \(\sin^3(x)\) as \(\sin^2(x) \cdot \sin(x)\). Write the integral with this change made.
(b) Now, we’ll convert \(\sin^2(x)\) to an expression that involves cosine instead. Use the identity \(\sin^2 \theta + \cos^2 \theta = 1\) to replace \(\sin^2(x)\) with something involving cosines, and write the integral with the change made.
(c) At this point, we can use a \(u\)-substitution to make this integral much easier. Substitute \(u = \cos(x)\). Write the new integral completely in terms of \(u\).
(d) Distribute any multiplication inside your integral, then integrate and complete the integral.
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