How do you integrate #int 1/sqrt(9x^218x+18) # using trigonometric substitution?
First of all, we complete the square/ re write in vertex form the quadratic under the square root:
So the integral can be re written as:
(Note the 9 under the square root has been factored out to the front)
Now substitute this into the integral to get:
Now evaluating the integral and reversing the substitution:
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To integrate ( \int \frac{1}{\sqrt{9x^2  18x + 18}} ) using trigonometric substitution, follow these steps:

Complete the square inside the square root to make it easier to work with. [ 9x^2  18x + 18 = 9(x^2  2x + 2) = 9[(x  1)^2 + 1] ]

Now, substitute ( x  1 = \sqrt{3}\tan(\theta) ), which implies ( x = 1 + \sqrt{3}\tan(\theta) ), and ( dx = \sqrt{3}\sec^2(\theta)d\theta ).

Substitute ( x ) and ( dx ) in terms of ( \theta ) in the integral. [ \int \frac{1}{\sqrt{9[(x  1)^2 + 1]}} dx = \int \frac{1}{\sqrt{9(1 + \tan^2(\theta))}} \cdot \sqrt{3}\sec^2(\theta) d\theta ]

Simplify inside the square root and outside the integral. [ \int \frac{\sqrt{3}\sec^2(\theta)}{\sqrt{9\sec^2(\theta)}} d\theta = \int \frac{\sqrt{3}\sec(\theta)}{3} d\theta ]

Now integrate ( \sqrt{3}\sec(\theta)/3 ) with respect to ( \theta ). [ \int \frac{\sqrt{3}\sec(\theta)}{3} d\theta = \frac{\sqrt{3}}{3} \int \sec(\theta) d\theta ]

Finally, integrate ( \sec(\theta) ) with respect to ( \theta ). [ \frac{\sqrt{3}}{3} \ln\sec(\theta) + \tan(\theta) + C ]

Substitute back ( \theta ) in terms of ( x ). [ \frac{\sqrt{3}}{3} \ln\sec(\arctan(\frac{x1}{\sqrt{3}})) + \tan(\arctan(\frac{x1}{\sqrt{3}})) + C ]

Simplify further if necessary.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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