In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ x^2 / √(x^2 - 4x + 5) dx

Verified step by step guidance

1. Start by examining the expression inside the square root: \(x^2 - 4x + 5\). To simplify this, complete the square for the quadratic expression.
2. Rewrite \(x^2 - 4x + 5\) as \((x^2 - 4x + 4) + 1\), which simplifies to \((x - 2)^2 + 1\).
3. Use the substitution \(u = x - 2\), which implies \(du = dx\). This changes the integral to an expression in terms of \(u\).
4. Rewrite the integral in terms of \(u\): replace \(x^2\) with \((u + 2)^2\) and \(\sqrt{x^2 - 4x + 5}\) with \(\sqrt{u^2 + 1}\). The integral becomes \(\int \frac{(u + 2)^2}{\sqrt{u^2 + 1}} \, du\).
5. Expand \((u + 2)^2\) to \(u^2 + 4u + 4\) and split the integral into three separate integrals: \(\int \frac{u^2}{\sqrt{u^2 + 1}} \, du + 4 \int \frac{u}{\sqrt{u^2 + 1}} \, du + 4 \int \frac{1}{\sqrt{u^2 + 1}} \, du\). Each of these can be found in standard integral tables.

Key Concepts

Substitution Method in Integration

The substitution method involves changing variables in an integral to simplify the integrand into a form that is easier to integrate. By choosing an appropriate substitution, often based on the inner function or expression, the integral can be transformed into a standard form found in integral tables.

Completing the Square

Completing the square is a technique used to rewrite quadratic expressions in the form ax^2 + bx + c as (x - h)^2 + k. This form simplifies the integrand, especially under square roots, making it easier to identify substitutions or match standard integral forms.

Using Integral Tables

Integral tables list standard integrals and their antiderivatives, providing quick references for common integral forms. After substitution and simplification, matching the integral to a form in the table allows for direct evaluation without performing integration from first principles.