Question

Evaluate the integral. $\int x \cos 5 x d x$

   Evaluate the integral.
$\int x \cos 5 x d x$

Calculus

James Stewart 7th Edition

Chapter 7, Problem 3

Instant Answer

Solved by Expert Danjoseph Quijada

12/22/2017

Step 1

Step 1: To evaluate the integral $\int x \cos 5x \, dx$, I'll use integration by parts with the formula: $\int u \, dv = uv - \int v \, du$ Show more…

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Evaluate the integral. $\int x \cos 5 x d x$

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Key Concepts

Integration by Parts

Integration by parts is a fundamental technique used to integrate products of functions. It is derived from the product rule of differentiation and allows one to transform a difficult integral into one that is easier to evaluate by selecting parts of the integrand as u (which is differentiated) and dv (which is integrated). This method is particularly useful when dealing with a product of polynomial and trigonometric, exponential, or logarithmic functions.

Substitution Method for Trigonometric Functions

The substitution method, often called u-substitution, is used to simplify the integration of composite functions. When integrating trigonometric functions with a linear argument, such as cosine of a linear expression, substitution helps adjust for the inner derivative, effectively reversing the chain rule. This method ensures that the integration process is aligned with the original function's structure and simplifies the evaluation of antiderivatives.

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