96. Challenge
Show that with the change of variables u = √tan x, the integral
∫ √tan x dx
can be converted to an integral amenable to partial fractions. Evaluate
∫[0 to π/4] √tan x dx.

Verified step by step guidance

1. Start with the substitution given: let \( u = \sqrt{\tan x} \). This means \( u^2 = \tan x \).
2. Differentiate both sides with respect to \( x \) to find \( du \) in terms of \( dx \): \( \frac{d}{dx}(u^2) = \frac{d}{dx}(\tan x) \) which gives \( 2u \frac{du}{dx} = \sec^2 x \). Solve for \( dx \) in terms of \( du \) and \( u \).
3. Express \( \sec^2 x \) in terms of \( u \) using the identity \( \tan^2 x + 1 = \sec^2 x \). Since \( \tan x = u^2 \), then \( \sec^2 x = 1 + u^4 \).
4. Rewrite the integral \( \int \sqrt{\tan x} \, dx = \int u \, dx \) by substituting \( dx \) from step 2 and \( \sec^2 x \) from step 3, to get the integral entirely in terms of \( u \) and \( du \).
5. Simplify the resulting integral to a rational function in \( u \) that can be decomposed into partial fractions. Then set up the partial fraction decomposition to prepare for integration.

Key Concepts

Change of Variables (Substitution)

This technique involves replacing the original variable with a new one to simplify the integral. By expressing the integral in terms of a new variable, complex expressions can become easier to handle, often transforming the integral into a more familiar or solvable form.

Partial Fraction Decomposition

Partial fractions break down a complex rational function into simpler fractions that are easier to integrate. This method is especially useful when the integrand is a rational expression, allowing the integral to be expressed as a sum of simpler terms.

Definite Integration with Trigonometric Limits

Evaluating definite integrals involving trigonometric functions requires careful substitution and adjustment of limits. When changing variables, the limits must be transformed accordingly to maintain the integral's value over the specified interval.