Question

which of the following techniques would you use to solve the integral \\(\int x^5 \sqrt{x^2 + 7} dx\\)?
that is, what is the best technique to apply to the above integral?
a. integration by parts with \\(dv = \sqrt{x^2 + 7}\\) and \\(u = x^5 dx\\).
b. u - substitution with \\(u = x^2 + 7\\).
c. integration by parts with \\(u = \sqrt{x^2 + 7}\\) and \\(dv = x^5 dx\\).
d. u - substitution with \\(u = x^5\\).
e. the integral is just \\(\frac{x^6}{9}(x^2 + 7)^{3/2} + c\\)
f. none of the above

Explanation:

To solve \(\int x^{5}\sqrt{x^{2}+7}dx\), we analyze each option:

• Option A: Integration by parts with \(dv = \sqrt{x^{2}+7}\) and \(u=x^{5}dx\) is incorrect because \(u\) should not have \(dx\) ( \(u\) is a function, not a differential) and this choice would complicate the integral further.
• Option B: Let \(u=x^{2}+7\), then \(du = 2xdx\) or \(xdx=\frac{du}{2}\). We can rewrite \(x^{5}\) as \(x^{4}\cdot x=(u - 7)^{2}\cdot x\). So \(x^{5}\sqrt{x^{2}+7}dx=(u - 7)^{2}\sqrt{u}\cdot\frac{du}{2}\), which is a polynomial in \(u\) times a root of \(u\), making the integral solvable by expanding the polynomial and integrating term - by - term. This is a valid u - substitution approach.
• Option C: Integration by parts with \(u=\sqrt{x^{2}+7}\) and \(dv = x^{5}dx\) would lead to \(v=\frac{x^{6}}{6}\) and \(du=\frac{x}{\sqrt{x^{2}+7}}dx\), and the new integral \(\int vdu=\int\frac{x^{6}}{6}\cdot\frac{x}{\sqrt{x^{2}+7}}dx\) is more complicated than the original, so this is not the best technique.
• Option D: If \(u = x^{5}\), then \(du=5x^{4}dx\), and the remaining part \(\sqrt{x^{2}+7}\) does not match the differential \(du\) in a way that simplifies the integral, so this substitution is not helpful.
• Option E: Differentiating \(\frac{x^{6}}{9}(x^{2}+7)^{\frac{3}{2}}+C\) using the product rule gives \(\frac{6x^{5}}{9}(x^{2}+7)^{\frac{3}{2}}+\frac{x^{6}}{9}\cdot\frac{3}{2}(x^{2}+7)^{\frac{1}{2}}\cdot2x=\frac{2x^{5}}{3}(x^{2}+7)^{\frac{3}{2}}+\frac{x^{7}}{3}(x^{2}+7)^{\frac{1}{2}}\), which is not equal to \(x^{5}\sqrt{x^{2}+7}\), so this is incorrect.

Answer:

B. u - substitution with \(u = x^{2}+7\)