# How do you integrate #int sqrt(-x^2-10x)/xdx# using trigonometric substitution?

Use the substitution

Let

Complete the square in the square root:

Simplify:

Integrate directly:

Reverse the substitution:

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The answer is

Complete the square :

Therefore, the integral is

Therefore,

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To integrate ( \int \frac{\sqrt{-x^2 - 10x}}{x} , dx ) using trigonometric substitution, let ( x = -5 \sin(\theta) - 5 ). Then, ( dx = -5 \cos(\theta) , d\theta ). Substitute these expressions into the integral and simplify it using trigonometric identities.

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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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