Question

score on last try: 0 of 1 pt. see details for more. at least one scored part is incorrect. jump to last submitted part. next question you can retry this question below consider the indefinite integral \\( \int x^{6} \cdot \sqrt{x^{7} - 7} dx \\): this can be transformed into a basic integral by letting \\( u = \\) \\( x^7 \\) or \\( x^7 - 7 \\) and \\( du = \square dx \\) performing the substitution yields the integral \\( \int \square du \\) question help: video

Explanation:

Step1: Choose substitution variable

We have the integral \(\int x^{6}\cdot\sqrt{x^{7}-7}dx\). Let's choose \(u = x^{7}-7\) because the derivative of \(x^{7}-7\) will relate to \(x^{6}\) which is present in the integrand.

Step2: Find derivative of u

Differentiate \(u = x^{7}-7\) with respect to \(x\). Using the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\), we get \(\frac{du}{dx}=7x^{6}\). Then, solving for \(dx\), we have \(du = 7x^{6}dx\), and \(\frac{du}{7}=x^{6}dx\).

Step3: Substitute into the integral

Now, substitute \(u = x^{7}-7\) and \(x^{6}dx=\frac{du}{7}\) into the original integral. The integral \(\int x^{6}\cdot\sqrt{x^{7}-7}dx\) becomes \(\int\sqrt{u}\cdot\frac{du}{7}=\frac{1}{7}\int u^{\frac{1}{2}}du\) (but for the purpose of the question, we just need to find the expression for \(du\) and the substituted integral form as per the blanks).

For the \(du\) part: From \(u=x^{7}-7\), \(du = 7x^{6}dx\)

For the substituted integral: \(\int\frac{\sqrt{u}}{7}du\) (or \(\frac{1}{7}\int u^{\frac{1}{2}}du\))

Answer:

• For \(u\): \(x^{7}-7\)
• For \(du\): \(7x^{6}\)
• For the integral after substitution: \(\frac{1}{7}\int\sqrt{u}du\) (or \(\frac{1}{7}\int u^{\frac{1}{2}}du\))