# Integration by Parts versus the Power Rule

• Mandelbroth
In summary: Essentially, the conversation discusses different methods for solving a calculus homework problem involving the integral of cos(ln x). One method involves integration by parts, while the other involves using the power rule for integration and the complex number definitions of trig functions. The conversation also touches upon the validity of using complex analysis in mathematical analysis. In summary, the conversation explores different approaches to solving a calculus problem involving the integral of cos(ln x) and discusses the interconnection between real and complex analysis in solving such problems.
Mandelbroth
Recently, a friend of mine asked for help on their calculus homework. The problem was to find $\int cos(ln \ x) \ dx$. However, I've never gotten around to memorizing the derivatives and integrals of the trig functions.

I know that you can do it using integration by parts, with $\int cos(ln \ x) \ dx = x \ cos(ln \ x) + \int sin(ln \ x) \ dx = x \ cos(ln \ x) + x \ sin(ln \ x) - \int cos(ln \ x) \ dx$, implying that $2\int cos(ln \ x) \ dx = x \ cos(ln \ x) + x \ sin(ln \ x)$, and thus that $\int cos(ln \ x) \ dx = \frac{x}{2}(cos(ln \ x) + sin(ln \ x)) + C$.

However, I used the power rule for integration (I think the technical name is Cavalieri's quadrature formula). $\int cos(ln \ x) \ dx = \int \frac{e^{i \ lnx}+e^{-i \ lnx}}{2} \ dx = \int \frac{x^{i}+x^{-i}}{2} \ dx = \frac{1}{2}(\frac{x^{i+1}}{i+1}+\frac{x^{1-i}}{1-i}) + C = \frac{x}{2}(\frac{x^{i}}{i+1}+\frac{x^{-i}}{1-i}) + C = \frac{x}{2}(\frac{x^{i}(1-i)+x^{-i}(1+i)}{2}) + C$, which is obviously equivalent to the previous answer.

Can I take it from this problem that the process of using the complex number definitions of the trig functions is valid for all such integrals? That is, does Cavalieri's quadrature formula ALWAYS work for complex numbers (aside from the obvious of x-1, which is why I am asking...)?

For any mathematical analysis (integration, etc.) using the exponential representation for trig. functions is always valid.

Indeed, there is great interconnection between real and complex analysis. In fact, there is a rigorous method to calculate real integrals using complex analysis (using Cauchy's residue theorem).

## 1. What is the difference between Integration by Parts and the Power Rule?

Integration by Parts and the Power Rule are two different methods used in calculus to find the integral of a function. Integration by Parts involves using the product rule of differentiation to break down an integral into simpler parts, while the Power Rule is a direct application of the power rule of differentiation.

## 2. When should I use Integration by Parts over the Power Rule?

Integration by Parts is typically used when the integral involves a product of two functions, while the Power Rule is used when the integral involves a function raised to a power. If the integral involves both a product and a power, it may be necessary to use both methods in combination.

## 3. Which method is more difficult to use?

Neither method is inherently more difficult, as they both involve different strategies for finding the integral. However, some integrals may be more easily solved using one method over the other, so it is important to be familiar with both methods and choose the one that is most appropriate for the given integral.

## 4. Can I use Integration by Parts and the Power Rule together?

Yes, it is possible to use both methods together in certain cases. For example, if the integral involves a product of two functions and one function is raised to a power, you may need to use Integration by Parts for the product and then the Power Rule for the power.

## 5. Are there any limitations to using Integration by Parts or the Power Rule?

Integration by Parts and the Power Rule are both powerful methods for finding integrals, but they do have their limitations. In some cases, the integral may not be solvable using these methods and other techniques, such as substitution or partial fractions, may need to be used instead.

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