Question

in using the technique of integration by parts, you must carefully choose which expression is u. for the following problem, use the guidelines in this section to choose u. do not evaluate the integral. \\(\int x^7 \sin(4x) dx\\)

Explanation:

Step1: Recall Integration by Parts Guideline

When using integration by parts \(\int u \, dv = uv - \int v \, du\), we typically choose \(u\) as a function that becomes simpler when differentiated (e.g., polynomials, logarithmic functions, inverse trigonometric functions) and \(dv\) as a function that is easy to integrate (e.g., exponential, trigonometric functions).

Step2: Analyze the Integrand

The integrand is \(x^{7}\sin(4x)dx\). Here, \(x^{7}\) is a polynomial (which becomes simpler when differentiated, as the power decreases) and \(\sin(4x)\) is a trigonometric function (easy to integrate). So we choose \(u = x^{7}\) (since differentiating \(x^{7}\) will reduce the power, making subsequent integrals simpler, while integrating \(\sin(4x)\) is straightforward).

Answer:

\(u = x^{7}\)