Calculus: Early Transcendentals
8th Edition
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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![The mathematical expression in the image is an integral:
\[
\int x^3 \cos x \, dx
\]
This is an example of an indefinite integral involving the product of a polynomial and a trigonometric function. Solving this integral typically involves integration by parts, a technique used to integrate products of functions.
To solve this integral using integration by parts, you can let \( u = x^3 \) and \( dv = \cos x \, dx \). Then, you differentiate and integrate these parts to find the solution:
1. \( u = x^3 \) implies \( du = 3x^2 \, dx \)
2. \( dv = \cos x \, dx \) implies \( v = \sin x \)
Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we get:
\[
\int x^3 \cos x \, dx = x^3 \sin x - \int 3x^2 \sin x \, dx
\]
This integral may further require repeated application of integration by parts or other integration techniques like tabular integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80321e65-f5af-4407-990e-2a73bdb503ea%2F3e307d26-9700-4045-b9d9-d200f1ec0254%2F8rsew6_processed.png&w=3840&q=75)
Transcribed Image Text:The mathematical expression in the image is an integral:
\[
\int x^3 \cos x \, dx
\]
This is an example of an indefinite integral involving the product of a polynomial and a trigonometric function. Solving this integral typically involves integration by parts, a technique used to integrate products of functions.
To solve this integral using integration by parts, you can let \( u = x^3 \) and \( dv = \cos x \, dx \). Then, you differentiate and integrate these parts to find the solution:
1. \( u = x^3 \) implies \( du = 3x^2 \, dx \)
2. \( dv = \cos x \, dx \) implies \( v = \sin x \)
Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we get:
\[
\int x^3 \cos x \, dx = x^3 \sin x - \int 3x^2 \sin x \, dx
\]
This integral may further require repeated application of integration by parts or other integration techniques like tabular integration.
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