Problem 26 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the indefinite integral. \(\int \frac{\cos (\pi / x)}{x^{2}} d x\)

Step-by-Step Solution

Answer

The indefinite integral is \(-\pi \sin(\pi / x) + C\).

Step 1: Identify the Substitution

To solve the integral \( \int \frac{\cos (\pi / x)}{x^{2}} \, dx \), we notice the expression \( \pi / x \). To simplify the integral, we'll use substitution. Let's set \( u = \pi / x \). Calculate the differential: we have \( x = \pi / u \), so \( dx = -\frac{\pi}{u^2} \, du \).

Step 2: Substitute and Transform the Integral

Substitute \( u = \pi / x \) and \( dx = -\frac{\pi}{u^2} \, du \) into the integral. The integral becomes:\[ \int \frac{\cos(u)}{(\pi/u)^2} \left(-\frac{\pi}{u^2}\right) du = -\pi \int \cos(u) \, du \]

Step 3: Integrate the Cosine Function

Integrate \(-\pi \int \cos(u) \, du\), which results in:\[-\pi (\sin(u)) + C\]where \( C \) is the constant of integration.

Step 4: Substitute Back the Original Variable

Re-substitute \( u = \pi / x \) back into the expression:\[-\pi \sin(\pi / x) + C\]This gives us the antiderivative in terms of \( x \).

Key Concepts

Substitution Method

The substitution method is a powerful technique in integration which is used to simplify integrals by changing variables. It's often referred to as "u-substitution," in which a substitution is made to transform the integral into a more straightforward form. In our exercise, the integral \( \int \frac{\cos(\pi / x)}{x^2} \, dx \) involves a trigonometric function where \( \pi / x \) is a complex expression. To simplify this, we let \( u = \pi / x \). Now, since \( x = \pi / u \), we must express \( dx \) in terms of \( u \), which results in the differential equation: \( dx = -\frac{\pi}{u^2} \, du \).What substitution does is change the variable of integration. It replaces the original variable \( x \) to \( u \), which makes the integral easier to evaluate. It is essential to also change \( dx \) to match the new variable \( du \), ensuring the integral remains accurately expressed.

Integration Techniques

Integration, much like differentiation, comes with a variety of techniques to aid in finding antiderivatives, or indefinite integrals. A few common techniques include:

• Substitution: As seen in our exercise, we use substitution when an integral contains a function and its derivative.
• Integration by Parts: Useful when the integral is a product of two functions, following the ILATE rule (Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential).
• Partial Fraction Decomposition: Suitable for rational functions where the degree of the polynomial in the denominator exceeds the numerator's degree.
• Trigonometric Identities: Helps simplify integrals involving trigonometric expressions, using identities like \( \sin^2(x) + \cos^2(x) = 1 \).

In our problem, the substitution method proved most helpful. After substitution, the integral transformed into \(-\pi \int \cos(u) \, du\), a much simpler form that can be integrated directly.

Antiderivatives

Antiderivatives, also known as indefinite integrals, are functions that "undo" differentiation. When you find the antiderivative of a function, you are essentially finding the original function prior to differentiation, plus a constant of integration because the derivative of a constant is zero.The antiderivative is expressed in the form \( \int f(x) \, dx = F(x) + C \), where \( C \) is any constant. This constant represents all possible vertical shifts of the antiderivative function.In our exercise, after performing the substitution and simplifying, we integrated \(-\pi \int \cos(u) \, du\) to find the antiderivative. The solution was \(-\pi \sin(u) + C\). Finally, substituting \( u = \pi / x \) back into the antiderivative gives us the final answer: \(-\pi \sin(\pi / x) + C\). This final expression is the antiderivative of the original integral in terms of \( x \).