Problem 26 - Essential Calculus

Question

Find the integral. $$ \int \frac{\sqrt{1-x}}{\sqrt{x}} d x $$

Step-by-Step Solution

Answer

The integral of \(\frac{\sqrt{1-x}}{\sqrt{x}}\) dx is \(2\ln|\sqrt{x}| - x\)

Step 1: Apply substitution

Let \( u = \sqrt{x} \). This implies that \( du = (1/2\sqrt{x}) dx \). Now substitute these values into the integral.

Step 2: Simplify the integrand

The integral now takes the form \( \int 2\frac{\sqrt{1-u^2}}{u} du \). This can be split into two separate integrals: \( \int 2\frac{1}{u} du - \int 2u du\)

Step 3: Solve the two integrals

The first integral can be solved as \(2 \ln |u|\) and the second integral as \(-u^2\).

Step 4: Substitute back

Substitute \(u= \sqrt{x}\) back into both solutions to get the final answer: \(2\ln|\sqrt{x}| - x\)