Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{\ln ^{6} x}{x} d x$$

Short answer

The evaluated integral of the function \(\frac{\ln^{6}x}{x}\) with respect to \(x\) is \(\frac{1}{7}(\ln x)^{7} + C\).

Step-by-step solution

Identify an appropriate substitution

Take \(u=\ln x\). Why this substitution is an adequate choice? It's because the derivative of \(\ln x\) is \(\frac{1}{x}\), which is present in the integral. So, when we substitute, the \(\frac{1}{x}\) in the denominator will cancel out.

Calculate the differential du

The differential \(du\) is the derivative of \(u\) with respect to \(x\), multiplied by \(dx\). Calculate \(du\) using the substitution \(u = \ln x\), which yields \(du = \frac{1}{x}\ dx\).

Substitute into the Integral

Substitute \(u\) and \(du\) into the original integral. The integral then becomes \(\int u^{6} du\). This is a much simpler integral to compute, as it involves a basic power rule.

Evaluate the Integral

The integral \(\int u^{6} du\) can be easily evaluated using the Power Rule for Integrals. The Power Rule states that the integral of \(u^n\) with respect to \(u\) is \(\frac{1}{n+1} u^{n+1}\) (plus a constant). Applying this rule, we get \(\frac{1}{7}u^{7} + C\).

Substitute Back for \(x\)

Substitute back in for \(x\) using the original substitution \(u = \ln x\). The final result is then \(\frac{1}{7}(\ln x)^{7} + C\).