Integration with Trigonometric Substitution

• janofano
In summary, the conversation is about solving the integral I=(x)sqrt(9-x^2)dx using trigonometric substitution. The solution is to use the substitution u=9-x^2 and du=-2xdx. However, the user points out that this is not necessary and offers an alternative solution. Another user suggests analyzing the given solution and practicing it, to which the first user responds that they have already done so. The conversation ends with a user pointing out an error in the first user's substitution and providing the correct substitution of u=x^2 and dx=3cos(u)du.
janofano
[SOLVED] Integration with Trigonometric Substitution

Homework Statement

Given integral (I):
I[(x)sqrt(9-x^2)dx]

by words:
Integral of "X" times square root of "9-X(squared)

Use proper trigonometric substitution to solve this problem.

The Attempt at a Solution

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You don't even need Trig substitution.

$$\int x\sqrt{9-x^2}dx$$

$$u=9-x^2$$
$$du=-2xdx \rightarrow xdx=-\frac 1 2 du$$

I know that I don't need that.
But the problem is, I have to use it.
The exercise require it.

Well you posted the solution to it? I don't know what else to tell you. Just analyze what they did. Work it yourself a couple times if you have to.

You forgot a term when you first did the substitution.

x = 3sin(u)
dx = 3cos(u)du

(9 - x^2)^(1/2) = 3cos(u)

So the integral becomes 27sin(u)cos^2(u)du

$$\int x^2\sqrt{9-x^2}dx$$
apropo
$$u=x^2\sqrt{9-x^2}dx$$

Last edited:

What is "Integration with Trigonometric Substitution"?

Integration with Trigonometric Substitution is a method of solving integrals that involve trigonometric functions by using a substitution technique. It is particularly useful for integrals containing functions like sine, cosine, and tangent.

When should I use Trigonometric Substitution for integration?

Trigonometric Substitution is most useful when the integrand contains a combination of algebraic and trigonometric functions, such as √(x²+1) or √(1-x²). It can also be used when the integrand contains expressions with radicals, such as √(a²-x²) or √(x²-a²).

How do I use Trigonometric Substitution for integration?

The general steps for using Trigonometric Substitution are:
1. Identify the type of substitution needed based on the form of the integrand.
2. Substitute the given variable with the corresponding trigonometric function.
3. Use trigonometric identities to simplify the integral.
4. Solve the new integral using standard integration techniques.
5. Substitute back the original variable to obtain the final answer.

What are the common trigonometric substitutions used in integration?

The most commonly used trigonometric substitutions are:
1. For integrals with expressions containing √(a²-x²), use x = a sinθ or x = a cosθ substitution.
2. For integrals with expressions containing √(x²+a²), use x = a tanθ or x = a secθ substitution.
3. For integrals with expressions containing √(x²-a²), use x = a secθ or x = a tanθ substitution.

Can Trigonometric Substitution always be used to solve integrals?

No, Trigonometric Substitution is not always applicable to solve integrals. It is only useful for certain types of integrals that involve trigonometric functions. For other integrals, different techniques such as integration by parts or partial fractions may need to be used.

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