Integrating product of trig functs with different args

• electric.avenue
In summary, we can integrate a product of two trigonometric functions with different arguments by using the formulas for sin(a+b) and sin(a-b) and setting up an integrand loop. By using partial integration, we can solve for the integral and obtain the final solution.
electric.avenue

Homework Statement

How do you integrate a product of two trigonometric functions of x, when the argument is different, ie:

what is the integral of

sin x cos 3x

I ought to know this, but don't seem to be able to do it.

The Attempt at a Solution

Use the formulas for sin(a+b) and sin(a-b):

sin(a+b) = sin a cos b + cos a sin b
sin(a-b) = sin a cos b - cos a sin b

so

sin(a+b) + sin(a-b) = 2 sin a cos b

you can solve this by the formula
choose one part as v
choose the other as du

{-integral sigh

{vdu=v*u -{dv*u

Note that transgalactic's proposal will enable you to find the answer by entering an integrand loop of finite length.

To see what I mean with an "integrand loop", I'll take an example:

We are to compute:
$$I=\int_{0}^{1}e^{x}\cos(x)dx$$
we use partial integration to write this as:
$$I=\int_{0}^{1}e^{x}\cos(x)dx=e^{x}\cos(x)\mid^{x=1}_{x=0}+\int_{0}^{1}e^{x}\sin(x)dx=(e^{x}\cos(x)+e^{x}\sin(x))\mid_{x=0}^{x=1}-\int_{0}^{1}e^{x}\cos(x)dx=(e^{x}\cos(x)+e^{x}\sin(x))\mid_{x=0}^{x=1}-I$$
Therefore, we have gained the equation for I:
$$I=(e^{x}\cos(x)+e^{x}\sin(x))\mid_{x=0}^{x=1}-I$$
which is easily solved for I.

Thanks for all the help and advice, everyone. I got an A in maths, btw.

What is the definition of a product of trigonometric functions?

A product of trigonometric functions is a mathematical expression that involves multiplying two or more trigonometric functions together. These functions are typically sine, cosine, tangent, cotangent, secant, and cosecant.

How do you integrate a product of trigonometric functions with different arguments?

To integrate a product of trigonometric functions with different arguments, you can use the trigonometric identities to simplify the expression. Then, you can use integration by parts or substitution to evaluate the integral.

What are the common mistakes made when integrating a product of trigonometric functions with different arguments?

Some common mistakes include not using the correct trigonometric identities, forgetting to apply the chain rule when using substitution, and making errors in simplifying the expression.

Can a product of trigonometric functions with different arguments be simplified?

Yes, a product of trigonometric functions with different arguments can often be simplified using trigonometric identities. This can make the integration process easier and more efficient.

What are some real-life applications of integrating products of trigonometric functions with different arguments?

Integrating products of trigonometric functions can be used in various fields such as physics, engineering, and economics. For example, in physics, it can be used to calculate the work done by a force in circular motion, and in economics, it can be used to model cyclical patterns in data.

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