How do you integrate #int 1/sqrt(x^2+2x)# by trigonometric substitution?
This is a known integral that can be derived here
We have one more substitution to reverse.
Hopefully this helps!
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To integrate ∫(1/√(x^2 + 2x)) dx using trigonometric substitution, we follow these steps:

Complete the square: Rewrite the denominator in the form (x + 1)^2  1. x^2 + 2x = (x^2 + 2x + 1)  1 = (x + 1)^2  1.

Apply the trigonometric substitution: Let x + 1 = √2 * tan(θ), then dx = √2 * sec^2(θ) dθ.

Substitute x + 1 = √2 * tan(θ) into the integral and express everything in terms of θ.

Solve for dx in terms of dθ.

Substitute dx in terms of dθ and simplify the integral.

Integrate the resulting expression with respect to θ.

Finally, express the result in terms of x by reversing the substitution used in step 2.
By following these steps, you can integrate ∫(1/√(x^2 + 2x)) dx using trigonometric substitution.
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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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