How to Solve the Integral ∫-1/(x√x^2-1) dx with Step-by-Step Explanation

∫-1/(x√x^2-1) dx

To solve the integral ∫-1/(x√x^2-1) dx, we can begin by making a u-substitution to simplify the expression

To solve the integral ∫-1/(x√x^2-1) dx, we can begin by making a u-substitution to simplify the expression.

Let’s let u = x^2 – 1. Taking the derivative of both sides with respect to x, we get du/dx = 2x.

Rearranging this equation, we have dx = du/(2x).

Now substituting u and dx in terms of x into the original integral, we get:

∫-1/(x√x^2-1) dx = ∫-1/(x√u) (du/2x)

Notice that the x term in the numerator cancels out with one of the x terms in the denominator, simplifying the integral to:

∫-1/(√u) (du/2)

We can continue to simplify the integral by moving constant terms out of the integral:

1/2 ∫-1/√u du

To integrate -1/√u, we need to perform another substitution. Let v = √u, so v^2 = u, and 2vdv = du.

Now, substituting v and dv into the integral, we get:

1/2 ∫-1/v (2vdv)

Simplifying further, we have:

-∫1 dv

Integrating -1 with respect to v, we get -v + C, where C is the constant of integration.

Substituting back for u and x, our final solution for the integral is:

-√u/2 + C

Remember that u = x^2 – 1, so the final answer is:

-√(x^2 – 1)/2 + C

Therefore, the integral of -1/(x√x^2-1) dx is -√(x^2 – 1)/2 + C, where C is the constant of integration.

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