Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\int \frac{d x}{\sqrt{x^{2}+4 x+5}}\)

Step-by-Step Solution

Answer

\( \int \frac{dx}{\sqrt{x^2+4x+5}} = \sinh^{-1}(x+2) + C \).

Step 1: Complete the Square

First, we need to complete the square for the expression under the square root in the denominator, \(x^2 + 4x + 5\).To complete the square, take the coefficient of \(x\) in \(4\), divide it by 2 giving us \(2\), and then square it, which gives us \(4\).Therefore, \(x^2 + 4x + 5 = (x+2)^2 + 1\).

Step 2: Substitution

Let's use a substitution to simplify the integration.Let \(u = x + 2\), which implies \(du = dx\) and the expression under the root becomes:\( \sqrt{(x+2)^2 + 1} = \sqrt{u^2 + 1} \). The integral becomes \( \int \frac{du}{\sqrt{u^2 + 1}} \).

Step 3: Identify the Integral Form

Recognize that the integral \( \int \frac{du}{\sqrt{u^2 + 1}} \) is a standard form that can be solved using an inverse hyperbolic function:\( \sinh^{-1} u + C \), where \(C\) is the constant of integration.

Step 4: Back-substitution

Substitute back the expression for \(u\):From our substitution, \(u = x + 2\), so \( \sinh^{-1} u = \sinh^{-1} (x + 2)\).Thus, the integral \( \int \frac{dx}{\sqrt{x^2 + 4x + 5}} = \sinh^{-1}(x+2) + C \), where \(C\) is the constant of integration.

Key Concepts

Completing the Square

When you are looking at expressions like the one in the given integral, \(x^2 + 4x + 5\), completing the square can transform it into an easier form. This is an algebraic technique to change a quadratic expression into a perfect square plus a constant. Not only does it help with integration, but it also simplifies solving equations and graphing parabolas.
Here's the step-by-step process:

• Identify the coefficient of \(x\) in the quadratic expression. In our case, the coefficient is 4.
• Divide this number by 2, which gives us 2.
• Square the result to get 4.

Now, incorporate this new number, so the expression \(x^2 + 4x + 5\) becomes \((x+2)^2 + 1\).
This transformation makes the expression under the square root a simple form to integrate in later steps by laying the foundation for using inverse hyperbolic functions.

Inverse Hyperbolic Functions

In integration, when you encounter forms like \int \frac{1}{\sqrt{u^2 + 1}} \, du\, inverse hyperbolic functions, specifically \(\sinh^{-1}\), become valuable tools. They serve as antiderivatives for certain expressions under the integral sign.
Here's a little more on the topic:

• Hyperbolic functions are analogues of trigonometric functions and are useful in a variety of calculus applications, including certain integrals.
• The inverse hyperbolic sine function, \( \sinh^{-1}(u)\), specifically allows us to integrate expressions involving square roots of quadratic polynomials.

So, in this exercise, when you identify the integral form \( \int \frac{du}{\sqrt{u^2 + 1}} \), it means that the solution can be expressed neatly as \( \sinh^{-1}(u) + C \), where \(C\) represents the integration constant.

Substitution Method

The substitution method is a powerful integration technique often used to simplify an integral into a recognizable form. This is done by making a substitution that changes variables, simplifying the expression we need to integrate.
In our example, we performed a substitution to make integration easier:

• We set \(u = x + 2\), aligning it with the perfect square form created from completing the square.
• This transforms the variable expressed in terms of \(x\) into \(u\), hence also expressing \(dx\) in terms of \(du\).

Now, the integral \int \frac{dx}{\sqrt{x^2 + 4x + 5}}\ is converted to the more manageable form \( \int \frac{du}{\sqrt{u^2 + 1}} \). This substitution elegantly sets up the problem for solving with inverse hyperbolic functions.
Substitution allows you to look at the problem from a different angle, simplifying calculations and helping recognize integral forms.