Question

\(\int \frac{x^{2}+2}{x^{4}+4} d x\) is equal to (1) \(\frac{1}{2} \tan ^{-1}\left(\frac{x^{2}+2}{2 x}\right)+c\) (2) \(\frac{1}{2} \tan ^{-1}\left(\frac{x^{2}-2}{2 x}\right) c\) (3) \(\frac{1}{2} \tan ^{-1}\left(\frac{2 x}{x^{2}-2}\right)+c\) (4) \(\frac{1}{2} \tan ^{-1}\left(x^{2}+2\right)+c\)

Short answer

Option 1\left(\frac{1}{2}\tan^{-1}\left(\frac{x^2+2}{2x}\right)+c\) is correct

Step-by-step solution

Simplify the integral

Notice that the numerator and the integral suggest a substitution involving the derivative of a fraction. To simplify, set an auxiliary function in the form \(u = x^2\). Then, \(du = 2x \, dx\). Thus, the integral becomes: \[ \frac{1}{2} \times \frac{1+2/x^2}{u^2+4} \times du \text{ since } dx = \frac{du}{2x} \]

Decompose the integral

Based on the substitution, our integral becomes:\[ \frac{1}{2} \times \frac{1+ 2u}{u^4+4} du \]

Further simplification

Now, consider recognizing that the denominator can be decomposed as a sum of squares. This step suggests breaking the integral into simpler or recognizable parts. Noticing a standard form for inverse tangent integral: \[ \frac{d}{du} \left(\tan^{-1} \left(\frac{u}{2} \right) \right)= \frac{2}{u^2+4} \]

Apply standard integral

Rewrite our integral in terms of this pattern: \[ \frac{1}{2} \times \frac{1}{2} \left( \frac{1}{ \frac{u^2}{2}+2}\right)5dt = \frac{1}{2]\]}

Key concepts

Definite Integral

An integral can be 'definite' or 'indefinite'. A definite integral has limits of integration where the value of the integral is determined within a specific range. Definite integrals are denoted as: \( \int_{a}^{b} f(x) \, dx \) This means you are integrating the function f(x) from x = a to x = b, providing the area under the curve between these two points. The limits a and b are called the 'lower' and 'upper' limits of the integral, respectively. The value of a definite integral can often be calculated using the Fundamental Theorem of Calculus. It states: \[ \int_{a}^{b} f(t) \, dt = F(b) - F(a) \] where F is the antiderivative of f, which means that F'(x) = f(x). Thus, finding a definite integral involves first finding an antiderivative, then evaluating it at the boundaries.

Substitution Method

The substitution method, also known as u-substitution, is a technique for finding integrals. It's particularly useful for integrals involving composite functions. The basic idea is to simplify the integrand by substituting a part of it with a single variable u. Here are the steps for substitution:

• Choose a substitution: Pick a part of the integral to substitute with u. Common substitutions involve selecting expressions inside powers, roots, or trigonometric functions.


• Differentiate: Compute du, which is the derivative of u with respect to x.


• Rewrite the integral: Replace the original variables in the integral with u and du.


• Integrate: Perform the integration with respect to u.


• Back-substitute: Convert the result back into the original variable x.

In our example, we chose u = x\( ^2 \). Then, du = 2x dx, which simplified the original integral into a form that's easier to handle.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the angle corresponding to a given trigonometric value. The most common ones are:

• \text {arcsin (or sin} \^{-1} )


• \text{arccos (or cos} \^{-1} )


• \text{arctan (or tan} \^{-1} )

Integral calculations often utilize the inverse tangent, arcsin, and arccos functions. For example, a standard integral result is: \[ \int \frac{du}{a^2 + u^2} = \frac{1}{a} \tan^{-1} \left( \frac{u}{a} \right) + C \]In our exercise, we used the standard form which resembles the arctangent function. Recognizing this pattern allowed us to manage the integral effectively, finally resulting in terms of \tan^{-1}/(inverse tangent).