Question

Integral of sqrt(x²-1)/x (Substitution)

A: Trigonometric Substitution |
B: Partial Fraction Decomposition |
C: Integration by Parts |
D: u-Substitution

Answer 1

To calculate the integral of sqrt(x²-1)/x, we can use the trigonometric substitution method. By substituting x = cos(u) and simplifying, the integral can be written as ln|sec(u) + tan(u)| - sin(u).

Step-by-step explanation:

In order to find the integral of sqrt(x²-1)/x, we can use the trigonometric substitution method.

Let x = cos(u), then dx = -sin(u) du. Substituting these values into the integral, we get:

∫(sqrt(cos²(u)-1)/cos(u)) * (-sin(u)) du = ∫(-sin²(u)/cos(u)) du

Using the trigonometric identity sin²(u) = 1 - cos²(u), we can rewrite the integral as:

∫((1-cos²(u))/cos(u)) du

Expanding the numerator and simplifying, we have:

∫(1/cos(u) - cos(u)) du

The first term is the integral of sec(u) du, which becomes ln|sec(u) + tan(u)|. The second term is the integral of -cos(u) du, which becomes -sin(u).

Therefore, the integral of sqrt(x²-1)/x can be simplified to ln|sec(u) + tan(u)| - sin(u), substituting back x = cos(u).