# Integration with Trig Sub: Simplifying the Square Root

• jhahler
In summary, the original problem is to find the integral of sqrt(x^2-36)/x). The student attempted to solve it using trig substitution, but made a mistake in substituting the value of 'a'. After correcting the mistake and simplifying the expression, the student will need to use the trig identity tan^2(u) = sec^2(u) - 1 to continue solving the integral. An easier substitution would be x=6sin(u).
jhahler

## Homework Statement

integral of sqrt(x^2-36)/x)

## Homework Equations

sqrt(x^2-a^2) = asec(u)
Pythagorean identity

## The Attempt at a Solution

I used trig sub on the x^2-36 and changed that to x=36sec(u) and dx= 36sec(u)tan(u). I simplified the square root in the numerator using sec^2(u)-1 = tan^2(u). This gave me (6tan(u)/36sec(u))(36sec(u)tan(u)) What trig identities do I need to proceed? Or did I use the wrong kind of trig sub? Any help is greatly appreciated in advance.

hi jhahler!
jhahler said:
This gave me (6tan(u)/36sec(u))(36sec(u)tan(u))

isn't that just tan2 ?

In your substitution, 'a' should be 6, not 36. Therefore your dx=6sec(u)tan(u).

After your change your numbers, and simplify, you will need to use the same trig identity you used in the beginning with tan^2(u).

I don't understand why you do such a complicated substitution. Perhaps I don't understand your notation right. It's way easier to use
$$x=6 \sin u.$$

jhahler said:
I used trig sub on the x^2-36 and changed that to x=36sec(u)
vanhees71 said:
… ##x=6 \sin u.## …

sec2 - 1 = tan2

sin2 - 1 = minus cos2

## What is integration with trigonometric substitution?

Integration with trigonometric substitution is a method used to solve integrals involving trigonometric functions. It involves substituting a trigonometric function for a variable in the integral, making it easier to solve.

## When is integration with trigonometric substitution used?

Integration with trigonometric substitution is commonly used when the integral contains expressions such as √(a^2 + x^2), √(a^2 - x^2), or √(x^2 - a^2). It can also be used when the integral contains expressions involving sin, cos, or tan raised to a power.

## How does integration with trigonometric substitution work?

The first step in integration with trigonometric substitution is to identify which trigonometric substitution to use. This is usually determined by the form of the integral. Next, the appropriate trigonometric function is substituted for the variable in the integral. The integral is then rewritten in terms of the trigonometric function and simplified. Finally, the integral is solved using basic integration techniques.

## What are the common trigonometric substitutions used in integration?

Some common trigonometric substitutions used in integration are x = a tanθ, x = a sinθ, x = a cosθ, and x = a secθ, where a is a constant and θ is a new variable.

## What are some tips for solving integrals with trigonometric substitution?

Some tips for solving integrals with trigonometric substitution include choosing the appropriate substitution, being familiar with trigonometric identities, and simplifying the integral as much as possible before attempting to solve it. It is also important to remember to substitute back in the original variable at the end to get the final answer.

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