Trig Identity Integral Homework: Solving a Tricky Equation Using Substitutions

• theRukus
In summary, the conversation discusses a problem involving trigonometric identities in integrals and suggests splitting the integral into two parts. It also mentions a differentiation formula that may be helpful in solving the problem.
theRukus

Homework Statement

I missed one class on trigonometric identities in integrals, and I feel that one is needed here:

$\int^3_0\frac{1+arctan(\frac{x}{3})}{9+x^2}dx$

The Attempt at a Solution

Again, I'm unsure what to do. I think that it is a trig identity, but I could be wrong. I'll continue to try parts & substitutions..

Thanks!

The derivative of acrtan(x) is 1/(x2+1), so the derivative of acrtan(x/3) is 3/(x2+9)

theRukus said:

Homework Statement

I missed one class on trigonometric identities in integrals, and I feel that one is needed here:

$\int^3_0\frac{1+arctan(\frac{x}{3})}{9+x^2}dx$

The Attempt at a Solution

Again, I'm unsure what to do. I think that it is a trig identity, but I could be wrong. I'll continue to try parts & substitutions..

Thanks!
Split the integral into two integrals. The first can be done directly and the second can be done with an ordinary substitution. Integration by parts is not the way to go.

Do you know this differentiation formula?
$$\frac{d}{dx} tan^{-1}(x)?$$

Notice that $\displaystyle \frac{d((f(x)^2)}{dx}=2f(x)f'(x)\,.$

That does that imply regarding $\displaystyle \int\ f(x)f\,'(x)\,dx\,?$

1. What is a trigonometric identity?

A trigonometric identity is an equation that relates different trigonometric functions, such as sine, cosine, tangent, etc. It is always true for all values of the variables involved.

2. What is the purpose of using trigonometric identities?

Trigonometric identities are used to simplify and manipulate trigonometric expressions, making them easier to solve or integrate. They also help in proving other mathematical theorems.

3. What is the difference between a trigonometric identity and a trigonometric equation?

A trigonometric identity is an equation that is always true, while a trigonometric equation is an equation that may or may not be true for certain values of the variables involved.

4. How do you prove a trigonometric identity?

To prove a trigonometric identity, you must start with one side of the equation and manipulate it using known trigonometric identities and algebraic properties until it matches the other side of the equation.

5. What are some common trigonometric identities?

Some common trigonometric identities include Pythagorean identities, double angle identities, half angle identities, and reciprocal identities. These identities can be used to simplify trigonometric expressions and solve equations involving trigonometric functions.

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