# Finding integrals of the product of trig functions

• DrWillVKN
In summary, the conversation discusses the difficulty of integrating exponential and trigonometric functions, specifically the integrals \intsin(a)*sin(b - a)da and \int e^{a} sin(a) da. Various methods are suggested, such as using trig identities and integration by parts. Ultimately, it is recommended to integrate the function \int e^{a} sin(a) da by parts twice and solve for the integral, or to use the trick of integrating \int e^{ix}dx to simplify the process.
DrWillVKN

## Homework Statement

I've come across integrals of exponential and trig functions and I have no idea how to do them. Integration by parts doesn't really work because they just derive into either e or another trig function.

One of them is $\int$sin(a)*sin(b - a)da
Another is $\int$e(a)*sin(a)da

## Homework Equations

sin(x + y) =sinx(cosy) + siny(cosx)
cos(x + y) = cosx(cosy) - sinx(siny)
sin^2(x) = (1 - cos2x)/2

## The Attempt at a Solution

I've tried to use the trig identity for sin(b-a), but that just gives an extremely long sin and cos statement that doesn't help. the one with e is even more confusing. How am I supposed to manually solve them?

Last edited:
For the second problem, let

$$I = \int e^{a} sin(a) da$$

Then integrate the right side by parts twice, and note that you get, as your integral, I. From there, just solve for I.

Another trick for integrating integrals like ∫exsin(x)dx is to instead do the integral ∫exeixdx =∫e(1+i)xdx as a simple exponential. Rationalize the result and note your original integral is the imaginary part. This avoids two integrations by parts and gives you the ∫excos(x)dx from the real part as a free bonus.

thanks for the replies, found the answers. i just didn't go far enough with the identities and integration by parts.

## 1. What are the basic steps for finding the integral of the product of trigonometric functions?

To find the integral of the product of trigonometric functions, you first need to identify the type of trigonometric functions involved (e.g. sine, cosine, tangent). Then, you can use trigonometric identities and integration techniques such as substitution or integration by parts to simplify the expression and find the integral.

## 2. Can I use any trigonometric identity to simplify the integral of the product of trig functions?

Yes, you can use any trigonometric identity that is relevant to the given expression. Some commonly used identities include the double angle, half angle, and sum/difference identities.

## 3. How do I know when to use substitution or integration by parts for finding the integral of the product of trig functions?

There is no set rule for when to use substitution or integration by parts. It often depends on the complexity of the expression and personal preference. However, a general guideline is to use substitution when there is a nested function (e.g. sin^2x) and integration by parts when there is a product of two functions.

## 4. Is there a specific way to approach finding the integral of the product of trig functions?

Yes, there are several approaches one can take to find the integral of the product of trig functions. Some common techniques include using trigonometric identities, simplifying the expression, and applying integration techniques such as substitution or integration by parts.

## 5. Are there any common mistakes to watch out for when finding the integral of the product of trig functions?

Yes, some common mistakes include forgetting to use the chain rule when using substitution, incorrectly applying trigonometric identities, and making errors when integrating by parts. It is important to carefully follow the steps and double check your work to avoid these mistakes.

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