Evaluate the integrals in Exercises 39–56.
55. ∫dx/(2√x + 2x)

Verified step by step guidance

1. Start by rewriting the integral to make it easier to work with. The integral is given as \(\int \frac{dx}{2\sqrt{x} + 2x}\). Factor out the common factor in the denominator: \(2(\sqrt{x} + x)\), so the integral becomes \(\int \frac{dx}{2(\sqrt{x} + x)}\).
2. Simplify the integral by factoring out the constant \(\frac{1}{2}\): \(\frac{1}{2} \int \frac{dx}{\sqrt{x} + x}\). This makes the integral \(\frac{1}{2} \int \frac{dx}{\sqrt{x} + x}\).
3. To handle the terms involving \(\sqrt{x}\) and \(x\), use the substitution \(t = \sqrt{x}\). Then, \(x = t^2\) and \(dx = 2t \, dt\). Substitute these into the integral to rewrite it in terms of \(t\).
4. After substitution, the integral becomes \(\frac{1}{2} \int \frac{2t \, dt}{t + t^2}\). Simplify the numerator and denominator inside the integral to get \(\int \frac{t}{t + t^2} \, dt\).
5. Simplify the integrand by factoring \(t\) in the denominator: \(t + t^2 = t(1 + t)\). Then the integrand becomes \(\frac{t}{t(1 + t)} = \frac{1}{1 + t}\). Now, the integral reduces to \(\int \frac{1}{1 + t} \, dt\), which is a standard integral.

Key Concepts

Integration of Rational Functions

This involves integrating functions expressed as ratios of polynomials or algebraic expressions. Recognizing how to simplify or manipulate the integrand into a more manageable form is essential for solving such integrals.

Substitution Method

A technique where a part of the integrand is replaced with a new variable to simplify the integral. It is especially useful when the integral contains composite functions or expressions involving roots and powers.

Simplifying Expressions with Radicals

Understanding how to rewrite and simplify expressions involving square roots or other radicals helps in making the integral easier to evaluate. This may include factoring, rationalizing, or expressing radicals as fractional exponents.