# Integral with trig substitution

• mbrmbrg
In summary, the conversation discusses an integration problem involving 6 times the integral of 1 over x squared minus x plus 1. The individual working on the problem uses substitution to simplify the integrand and arrives at an answer that is 6 times greater than the answer calculated by an integrator program. It is suggested that the discrepancy may be due to the missing 6 in the integrand entered into the program.
mbrmbrg
The bit of the problem that I'm working on:
$$6\int\frac{dx}{x^2-x+1}$$

My work:
$$=6\int\frac{dx}{(x^2-x+\frac{1}{4})+1-\frac{3}{4}}$$
$$=6\int\frac{dx}{(x-\frac{1}{2})^2+\sqrt{\frac{3}{4}}^2}$$

let $$x-\frac{1}{2}=\sqrt{\frac{3}{4}}\tan\theta$$
so $$dx=\sqrt{\frac{3}{4}}\sec^2\theta d\theta$$

$$=6\int\frac{\sqrt{\frac{3}{4}}sec^2\theta d\theta}{\frac{3}{4}+\frac{3}{4}tan^2\theta}$$
$$=(6)(\frac{\sqrt{3}}{2})(\frac{4}{3})\int\frac{sec^2\theta d\theta}{1+tan^2\theta}$$
$$=4\sqrt{3}\int d\theta$$
$$=4\sqrt{3}\theta$$

$$=4\sqrt{3}\arctan{\frac{2x-1}{\sqrt{3}}}$$

$$\frac{2}{\sqrt{3}}\arctan{\frac{2x-1}{\sqrt{3}}}$$

I don't think the two answers differ by a constant, but I can't find my error.

Your answer is 6 times greater than the integrator's. I assume you just didn't include the 6 when you put your integrand into the integrator since you can't write stuff before the integral when using that program. Try using the integrator again, bringing the 6 into the numerator of the integrand.

Thank you.

## What is "integral with trig substitution"?

"Integral with trig substitution" is a technique used in calculus to solve integrals that involve trigonometric functions. It involves substituting a trigonometric expression for a variable in the integral to make it more easily solvable.

## Why is trig substitution used?

Trig substitution is used because it can simplify the integral and make it easier to solve. It also allows for the use of trig identities and formulas to solve the integral.

## What are the steps for using trig substitution to solve an integral?

The steps for using trig substitution are:
1. Identify the appropriate trig substitution to use based on the form of the integral.
2. Substitute the trigonometric expression for the variable in the integral.
3. Use trig identities and formulas to simplify the integral.
4. Solve the resulting integral using basic integration techniques.
5. Substitute the original variable back into the solution to get the final answer.

## What are some common trig substitutions used in integrals?

Some common trig substitutions used in integrals include:
- Substituting x = sinθ or x = cosθ for integrals involving √(a^2 - x^2) or √(x^2 + a^2), respectively.
- Substituting x = tanθ for integrals involving √(x^2 + a^2) and x^2 + a^2.
- Substituting x = secθ or x = cscθ for integrals involving √(x^2 - a^2) or √(a^2 - x^2), respectively.

## Are there any restrictions when using trig substitution?

Yes, there are some restrictions when using trig substitution. The most common restriction is when the integral involves a radical and the variable in the radical is squared. In this case, the substitution may not work and a different method may need to be used. Additionally, the substitution may not work when the integral involves a rational function.

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