Problem 24 Integrate each of the given func... [FREE SOLUTION]

Chapter 28: Problem 24

Integrate each of the given functions. $$\int_{-4}^{0} \frac{d x}{x^{2}+4 x+5}$$

Short Answer

Expert verified

The integral evaluates to $2\tan^{-1}(2)$.

Step by step solution

Recognize the Denominator Form

Notice that the denominator can be rewritten in a standard quadratic form: $x^2 + 4x + 5$. Let's first check if this can be factored, but it doesn't factor neatly due to no real roots (discriminant). Thus, complete the square instead.

Complete the Square

Rewrite $x^2 + 4x + 5$ by completing the square. Take half of the coefficient of $x$, square it, and add and subtract: \[x^2 + 4x + 4 + 1 = (x + 2)^2 + 1.\] The integral now becomes $\int \frac{dx}{(x+2)^2 + 1}$.

Identify Standard Integral Form

Recognize that $\int \frac{dx}{(x+2)^2 + 1}$ resembles the arctangent integral formula $\int \frac{1}{a^2 + x^2}dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$. Here $a = 1$ and substitute $u = x+2$.

Change of Variables

Substitute $u = x + 2$, then $du = dx$, and change the limits of integration. When $x = -4$, $u = -2$, and when $x = 0$, $u = 2$. The integral becomes $\int_{-2}^{2}\frac{du}{u^2+1}$.

Integrate Using Arctangent Formula

Use the arctangent formula: $\int \frac{du}{u^2 + 1} = \tan^{-1}(u) + C$. So the integral $\int_{-2}^{2} \frac{du}{u^2 + 1}$ becomes $\left[\tan^{-1}(u)\right]_{-2}^{2}$.

Evaluate the Antiderivative at the Bounds

Evaluate $\tan^{-1}(u)$ from -2 to 2: $\tan^{-1}(2) - \tan^{-1}(-2)$. Since $\tan^{-1}(-x) = -\tan^{-1}(x)$, this is $\tan^{-1}(2) + \tan^{-1}(2) = 2\tan^{-1}(2)$.

Compute the Final Result

Calculate $2\tan^{-1}(2)$ to find the final result of the integral. Since $\tan^{-1}(2)$ is a known angle, it can be left in this form or approximated with a calculator for numerical answer if needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Completing the Square

Completing the square is a powerful algebraic technique that helps simplify quadratic expressions. Here's how it works: for a given quadratic in the form of $ax^2 + bx + c$, you can rewrite it as $(x - h)^2 + k$. The steps are quite simple and crucial for tackling integration problems that involve quadratic expressions.

First, focus on the $x^2$ and $x$ terms, effectively ignoring the constant initially.
Take half of the coefficient of $x$, square it, and add and subtract this square within the expression.
Factor the perfect square trinomial while adjusting the constants accordingly.

In the original problem, we completed the square for the quadratic $x^2 + 4x + 5$ to get $(x + 2)^2 + 1$. This simplifies the process of integration, especially when aiming to use standard trigonometric integral forms.

Arctangent Integral

The form $\int \frac{dx}{a^2 + x^2}$ is a well-known integral that simplifies to the arctangent function: $\frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$. Recognizing this pattern can save a lot of time and effort in calculus problems. Here's why this form is useful:

Combining algebraic rearrangement (such as completing the square) helps identify this integral form.
The arctangent formula provides a straightforward solution after rewriting the integral.
This specific type of integral appears often in calculus, arising from various applications, such as in geometry and physics.

In our example, the integral $\int \frac{dx}{(x+2)^2 + 1}$ matches the arctangent form with $a = 1$. This recognition allows you to apply the formula directly, speeding up the entire integration process.

Change of Variables

Change of variables, often called substitution, is a fundamental technique in integration. It simplifies the integral by transforming it into a more familiar form. Here's the basic idea:

Identify a substitution that will make the integral easier to solve, often converting a complicated expression into a simpler standard integral form.
Calculate the differential of your substitution to replace the variables in the integral.
Adjust the integration limits when dealing with definite integrals to match the new variable.

In the exercise, by substituting $u = x + 2$, the problem transitions to $\int \frac{du}{u^2 + 1}$. This formal change not only matches a known integral form (like arctangent) but also makes it simpler to evaluate within new integration limits. It's a strategy that makes solving complex integrals manageable and methodical.

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91影视

Integrate each of the given functions. $$\int_{-4}^{0} \frac{d x}{x^{2}+4 x+5}$$

Short Answer

Step by step solution

Recognize the Denominator Form

Complete the Square

Identify Standard Integral Form

Change of Variables

Integrate Using Arctangent Formula

Evaluate the Antiderivative at the Bounds

Compute the Final Result

Key Concepts

Completing the Square

Arctangent Integral

Change of Variables

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