59. Two Methods
b. Evaluate ∫(x / √(x + 1)) dx using substitution.

Verified step by step guidance

1. Identify the integral to solve: \(\int \frac{x}{\sqrt{x + 1}} \, dx\).
2. Choose a substitution to simplify the integral. Let \(u = x + 1\), so that \(du = dx\) and \(x = u - 1\).
3. Rewrite the integral in terms of \(u\): replace \(x\) with \(u - 1\) and \(dx\) with \(du\), giving \(\int \frac{u - 1}{\sqrt{u}} \, du\).
4. Split the integral into two simpler integrals: \(\int \frac{u}{\sqrt{u}} \, du - \int \frac{1}{\sqrt{u}} \, du\), which simplifies to \(\int u^{1/2} \, du - \int u^{-1/2} \, du\).
5. Integrate each term using the power rule for integrals: \(\int u^{n} \, du = \frac{u^{n+1}}{n+1} + C\), then substitute back \(u = x + 1\) to express the answer in terms of \(x\).

Key Concepts

Integration by Substitution

Integration by substitution is a technique used to simplify integrals by changing variables. It involves choosing a substitution that transforms the integral into a simpler form, often by letting a part of the integrand equal a new variable. This method is especially useful when the integral contains a composite function.

Algebraic Manipulation of Integrands

Before applying substitution, it is often necessary to rewrite the integrand in a form that makes substitution straightforward. This may involve factoring, expanding, or expressing parts of the integrand in terms of the substitution variable. Proper manipulation helps in identifying the correct substitution and simplifies the integration process.

Definite and Indefinite Integrals

Understanding the difference between definite and indefinite integrals is crucial. An indefinite integral represents a family of functions plus a constant of integration, while a definite integral computes the area under a curve between two limits. In substitution, adjusting the limits or adding the constant is important depending on the integral type.