# Using Trig Substitution in Trig Integration

• Zack K
In summary, the method for solving the given integral involves transforming it into ##-\frac{1}{4}\cot(\arcsin(\frac{1}{2}x))##, which can then be simplified using trigonometric ratios. The exact values of the opposite, adjacent, and hypotenuse do not matter, only their ratios. In LateX, it is recommended to use the commands "\cot" and "\arcsin" for better formatting.
Zack K

## Homework Statement

Integrate: $$\int \frac{dx}{x^2\sqrt{4-x^2}}dx$$

## The Attempt at a Solution

I got to the final solution ##\int \frac{dx}{x^2\sqrt{4-x^2}}dx=-\frac{1}{4}cot(arcsin(\frac{1}{2}x))##. But It's the method where you transform that to the solution ##-\frac{1}{4}cot(arcsin(\frac{1}{2}x))=-\frac{\sqrt{4-x^2}}{x}+C## that confuses me. I understand that you get that by seeing that on a right triangle, ##cot=\frac{adjacent}{opposite}##, but how do you know the value of the opposite, adjacent and hypotenuse?

Last edited:
Zack K said:
but how do you know the value of the opposite, adjacent and hypotenuse?
The exact values do not matter, just the ratios. You can choose the hypothenuse to have length one. Then you get the opposite because you know what the sine of the angle is. Pythagoras’ theorem then gives you the adjacent.

Zack K
Zack K said:

## Homework Statement

Integrate: $$\int \frac{dx}{x^2\sqrt{4-x^2}}dx$$

## The Attempt at a Solution

I got to the final solution ##\int \frac{dx}{x^2\sqrt{4-x^2}}dx=-\frac{1}{4}cot(arcsin(\frac{1}{2}x))##. But It's the method where you transform that to the solution ##-\frac{1}{4}cot(arcsin(\frac{1}{2}x))=-\frac{\sqrt{4-x^2}}{x}+C## that confuses me. I understand that you get that by seeing that on a right triangle, ##cot=\frac{adjacent}{opposite}##, but how do you know the value of the opposite, adjacent and hypotenuse?

Well, ##\cot(\arcsin(\frac1 2 x)) = \cos(\theta)/\sin(\theta),## where ##\sin \theta = \frac 1 2 x.##

BTW: in LateX, please write ##\cot ...## instead of ##cot ...##, and ##\arcsin ... ## instead of ##arcsin ...##. You do that by typing "\cot" instead of "cot", etc. (Similarly for all the other trig functions, the hyperbolic functions, and things like "ln", "log", "lim", "max", "min" and a host of others).

##dx## has got in there twice.

## What is trig substitution in trigonometric integration?

Trig substitution is a technique used in calculus to simplify the integration of trigonometric functions. It involves replacing a trigonometric expression with a single variable, usually denoted as t, and then using trigonometric identities to rewrite the expression in terms of t. This allows for easier integration using traditional integration techniques.

## When should I use trig substitution in trigonometric integration?

Trig substitution is typically used when the integral contains a combination of trigonometric functions, such as sin, cos, tan, sec, or csc. It can also be useful when the integral contains a radical expression or a quadratic expression.

## How do I choose the appropriate substitution for trigonometric integration?

Choosing the appropriate substitution for trigonometric integration depends on the form of the integral. Some common substitutions include using sin or cos when the integral contains an expression of the form a^2 - x^2, using tan when the integral contains an expression of the form a^2 + x^2, and using sec when the integral contains an expression of the form x^2 - a^2.

## What are some common trigonometric identities used in trig substitution?

Some common trigonometric identities used in trig substitution include the Pythagorean identities, such as sin^2x + cos^2x = 1, as well as the double angle identities, such as sin2x = 2sinx cosx and cos2x = cos^2x - sin^2x. These identities can be used to rewrite the integral in terms of the chosen substitution variable.

## Are there any tips for using trig substitution in trigonometric integration?

One helpful tip for using trig substitution is to always check your substitution by differentiating the resulting expression. If the derivative matches the original integral, then you have chosen the correct substitution. It is also important to be familiar with common trigonometric identities and to practice using them to simplify integrals.

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