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Chapter 7: Problem 47

Use integration by parts to derive the following reduction formulas. $$\int \ln ^{n} x d x=x \ln ^{n} x-n \int \ln ^{n-1} x d x$$

Short Answer

Expert verified
Question: Derive the reduction formula for the function $$\int \ln^n x dx$$, where n is a positive integer. Answer: The reduction formula for the given function is $$\int \ln^n x dx = x \ln^n x - n \int \ln^{n-1} x dx$$.

Step by step solution

01

Identify the functions u and dv

The given function is: $$\int \ln^n x dx$$ Let \(u = \ln^n x\) and \(dv = dx\).
02

Differentiate u to obtain du and integrate dv to obtain v

To differentiate u, we use the chain rule, where \(y = \ln x\). $$\frac{d}{dx} \ln^n x = n \ln^{n-1} x \frac{d}{dx} (\ln x) = \frac{n}{x} \ln^{n-1} x$$ So, \(du = \frac{n}{x} \ln^{n-1} x dx\). As for dv, we can integrate it directly to obtain v: $$v = \int dx = x$$
03

Apply the integration by parts formula

Substitute u, du, and v into the integration by parts formula: $$\int \ln^n x dx = x \ln^n x - \int x \frac{n}{x} \ln^{n-1} x dx$$
04

Combine the terms to derive the reduction formula

Simplify the equation obtained from Step 3: $$\int \ln^n x dx = x \ln^n x - n \int \ln^{n-1} x dx$$ Thus, we have derived the reduction formula: $$\int \ln^n x dx = x \ln^n x - n \int \ln^{n-1} x dx$$

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Most popular questions from this chapter

Consider the family of functions \(f(x)=1 / x^{p},\) where \(p\) is a real number. For what values of \(p\) does the integral \(\int_{0}^{1} f(x) d x\) exist? What is its value?

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