Evaluating the Integral of (du)/(u*sqrt(5 - u^2))

When evaluating integrals, it is important to identify the appropriate techniques and strategies to arrive at the solution. In this case, we will be evaluating the integral of (du)/(u*sqrt(5 - u^2)).

To begin, let's define the integral:

∫ (du)/(u*sqrt(5 - u^2))

To solve this integral, we will use the substitution method. Let's set u = sqrt(5) * sin(θ), where dθ = du / (sqrt(5) * cos(θ)).

Substituting this into the original integral, we get:

∫ (du)/(u*sqrt(5 - u^2)) = ∫ (dθ)/(sqrt(5) * cos(θ))

Now, let's simplify the expression further:

∫ (dθ)/(sqrt(5) * cos(θ)) = (1/sqrt(5)) * ∫ (dθ)/cos(θ)

Using the property of the natural logarithm, we can rewrite this as:

(1/sqrt(5)) * ∫ (dθ)/cos(θ) = (1/sqrt(5)) * ln(|sec(θ) + tan(θ)|) + C

Substituting back the original variable u, we get:

∫ (du)/(u*sqrt(5 - u^2)) = (1/sqrt(5)) * ln(|sec(arcsin(u/sqrt(5))) + tan(arcsin(u/sqrt(5)))|) + C

Remember, the absolute value is necessary when dealing with the natural logarithm to ensure that the argument is always positive.

In conclusion, the solution to the integral of (du)/(u*sqrt(5 - u^2)) is:

∫ (du)/(u*sqrt(5 - u^2)) = (1/sqrt(5)) * ln(|sec(arcsin(u/sqrt(5))) + tan(arcsin(u/sqrt(5)))|) + C

where C is the constant of integration.