# What is #int x/ sqrt(x^2 - 8^2) dx#?

# int x/(sqrt(x^2 - 64)) "d"x = sqrt(x^2 - 64) + C#

where

Here, we'll apply integration by substitution.

We can begin substituting and solving the integral since, as you can see, the substitution appears to fit our integral precisely:

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The integral of ( \frac{x}{\sqrt{x^2 - 8^2}} ) with respect to ( x ) can be evaluated using a trigonometric substitution. Let ( x = 8 \sec(\theta) ), then ( dx = 8 \sec(\theta) \tan(\theta) d\theta ). Substituting these into the integral gives:

[ \int \frac{x}{\sqrt{x^2 - 8^2}} dx = \int \frac{8 \sec(\theta)}{\sqrt{64\sec^2(\theta) - 64}} (8 \sec(\theta) \tan(\theta)) d\theta ]

Simplify the expression under the square root:

[ \sqrt{64\sec^2(\theta) - 64} = \sqrt{64(\sec^2(\theta) - 1)} = \sqrt{64 \tan^2(\theta)} = 8|\tan(\theta)| ]

Substitute back into the integral:

[ \int \frac{x}{\sqrt{x^2 - 8^2}} dx = \int \frac{8 \sec(\theta)}{8|\tan(\theta)|} (8 \sec(\theta) \tan(\theta)) d\theta = \int 64 d\theta ]

Integrating ( 64 ) with respect to ( \theta ) gives ( 64\theta + C ), where ( C ) is the constant of integration. Substitute ( \theta ) back in terms of ( x ) to get the final answer.

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The integral of ( \frac{x}{\sqrt{x^2 - 8^2}} ) with respect to ( x ) can be evaluated using a trigonometric substitution. Let ( x = 8\sec(\theta) ). Then, ( dx = 8\sec(\theta)\tan(\theta) , d\theta ). Substituting these values, the integral becomes:

[ \int \frac{8\sec(\theta)}{\sqrt{64\sec^2(\theta) - 64}} \cdot 8\sec(\theta)\tan(\theta) , d\theta ]

[ = \int \frac{64\sec^2(\theta)\tan(\theta)}{8|\sec(\theta)|} , d\theta ]

[ = 8\int |\tan(\theta)| , d\theta ]

[ = 8\int \tan(\theta) , d\theta ]

[ = -8\ln|\cos(\theta)| + C ]

[ = -8\ln|\cos(\arccos(\frac{x}{8}))| + C ]

[ = -8\ln\left|\frac{x}{8}\right| + C ]

[ = -8\ln|x| + 8\ln|8| + C ]

[ = -8\ln|x| + 8\ln|8| + C ]

[ = -8\ln|x| + 8\ln|8| + C ]

[ = -8\ln|x| + C ]

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When evaluating a one-sided 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 one-sided 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 one-sided 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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