Problem 63 - AP Calculus with 8 Practice Tests 15th

Question

\(\int \frac{2 x+1}{4+x^{2}} d x=\) (A) \(\ln \left(x^{2}+4\right)+C\) (B) \(\ln \left(x^{2}+4\right)+\tan ^{-1} \frac{x}{2}+C\) (C) \(\ln \left(x^{2}+4\right)+\frac{1}{2} \tan ^{-1} \frac{x}{2}+C\) (D) \(\ln \left(x^{2}+4\right)+\frac{1}{4} \tan ^{-1} x+C\)

Step-by-Step Solution

Answer

The answer is (C): \(\ln(x^2+4)+\frac{1}{2}\tan^{-1}\left(\frac{x}{2}\right)+C\).

Step 1: Identify the Integral Form

The given integral is \( \int \frac{2x+1}{4+x^2} \, dx \). This is a rational function where the degree of the numerator is less than that of the denominator. The structure of the integral suggests a combination of natural logarithm and inverse trigonometric functions.

Step 2: Decompose the Numerator

Notice that the numerator \(2x + 1\) can be expressed as two separate parts: a derivative of the denominator plus a constant. We write \(2x = d(4+x^2)/dx - 1\), which simplifies to \(2x+1 = 1 \cdot (4+x^2)' + 1\). This is useful for substitution.

Step 3: Split the Integral

Using the decomposition from Step 2, split the integral into two parts: \(\int \frac{2x}{4+x^2} \, dx + \int \frac{1}{4+x^2} \, dx\). We will solve each integral separately.

Step 4: Solve the First Integral

For \(\int \frac{2x}{4+x^2} \, dx\), use the substitution \(u = 4 + x^2\), which gives \(du = 2x \, dx\). This simplifies the integral to \(\int \frac{1}{u} \, du = \ln |u| = \ln(4 + x^2)\).

Step 5: Solve the Second Integral

The second integral \(\int \frac{1}{4+x^2} \, dx\) resembles the form of \(\int \frac{1}{a^2+x^2} \, dx\), which results in \(\frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right)\) with \(a=2\). Thus, it equals \(\frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right)\).

Step 6: Combine the Results

Add the results of the two integrals obtained in Steps 4 and 5. So, the integral \(\int \frac{2x+1}{4+x^2} \, dx\) becomes \(\ln(4+x^2) + \frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right) + C\), which matches choice (C).

Key Concepts

Inverse Trigonometric Integrals

Inverse trigonometric integrals often arise in calculus when dealing with expressions involving a sum of squares in the denominator. They are useful because many integrals involving trigonometric functions can be solved by leveraging inverse trigonometric identities. For example:

• The integral \( \int \frac{1}{a^2 + x^2} \, dx \) can be expressed as \( \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \). This identity is critical when handling integrals that fit the pattern \( \int \frac{1}{4+x^2} \) as seen in the original step-by-step solution.

Recognizing these forms helps in converting the integrals into simpler, often inverse trigonometric forms, allowing for straightforward evaluation. This approach simplifies complex integrals by transforming them into familiar antiderivatives, making them easier to resolve. It's essential to practice identifying these patterns quickly, as it greatly streamlines the integration process.

Rational Function Decomposition

Rational function decomposition plays a crucial role when the integral features a polynomial in the numerator and the denominator. Rational functions are expressed as the ratio of two polynomials, and if correctly decomposed, they simplify integration.

In the problem at hand, the numerator \( 2x + 1 \) is carefully decomposed into \( 2x = \frac{d}{dx}(4 + x^2) - 1 \). This insight allows the function to be split into simpler fractions.

• The decomposition benefits integration because it allows separate treatment of terms through substitution. The term \( 2x \) becomes part of a derivative in the related integral \( \int \frac{2x}{4+x^2} \, dx \), which can be simplified using substitutions to ease calculations.

Ultimately, decomposition reduces intricate rational expressions into manageable components, simplifying the integral to basic logarithmic and trigonometric forms.

Integral Substitution Methods

Substitution is a powerful technique to simplify integrals, especially when facing complex expressions. It involves replacing parts of the integral with a single variable, usually represented by \( u \), allowing the integral to transform into an easier form.

In this exercise, integral substitution was vital for solving \( \int \frac{2x}{4+x^2} \, dx \). By setting \( u = 4 + x^2 \), the differential \( du = 2x \, dx \) comes directly from the substitution, transforming the integral into \( \int \frac{1}{u} \, du = \ln|u| \). This substitution cleverly reduces the problem to a more straightforward logarithmic integral.

• This method streamlines the calculation process, particularly when the integral fits a known form, as it replaces complex integrands, easing the overall solving process. As seen in the exercise, handling \( \int \frac{1}{u} \, du \) becomes trivial after the substitution.

Understanding substitution applications increases efficiency in calculus, as it aligns complex integrals with standard forms, aiding in quicker solutions.