Problem 41 - Calculus Single Variable

Question

Evaluate the definite integrals. $$ \int_{0}^{1 / 2} x \tan \left(\pi x^{2}\right) d x $$

Step-by-Step Solution

Answer

The value of the integral is \( \frac{1}{4\pi} \ln(2) \).

Step 1: Substitution Selection

Choose an appropriate substitution to simplify the integral. Let \( u = \pi x^2 \). Then, \( du = 2\pi x \, dx \) or \( x \, dx = \frac{1}{2\pi} \, du \).

Step 2: Change of Limits

Determine the new limits of integration for \( u \) using the substitution \( u = \pi x^2 \). When \( x = 0 \), \( u = \pi(0)^2 = 0 \). When \( x = \frac{1}{2} \), \( u = \pi \left( \frac{1}{2} \right)^2 = \frac{\pi}{4} \).

Step 3: Substitute and Simplify the Integral

Substitute \( u \) and \( x \, dx \) in the integral. The integral becomes: \[ \int_{0}^{\pi/4} \frac{1}{2\pi} \tan(u) \, du \].

Step 4: Integrate the Function

Integrate \( \tan(u) \) to find the antiderivative. The antiderivative of \( \tan(u) \) is \( -\ln |\cos(u)| \). So the integral is:\[ \int_{0}^{\pi/4} \frac{1}{2\pi} \tan(u) \, du = \frac{1}{2\pi} [-\ln |\cos(u)|]_{0}^{\pi/4} \].

Step 5: Evaluate the Antiderivative at Both Limits

Calculate the values at the upper and lower limits:\[ -\ln |\cos(\pi/4)| - \left( -\ln |\cos(0)| \right) = -\ln \left( \frac{1}{\sqrt{2}} \right) + \ln(1) \].

Step 6: Simplify the Expression

Further simplify the expression:\[-\ln \left( \frac{1}{\sqrt{2}} \right) = \ln(\sqrt{2}) \] because \( \ln(a/b) = \ln(a) - \ln(b) \) and \( \ln(1) = 0 \).

Step 7: Final Result

Multiply by the constant \( \frac{1}{2\pi} \):\[ \frac{1}{2\pi} \ln(\sqrt{2}) = \frac{1}{4\pi} \ln(2) \] since \( \ln(\sqrt{2}) = \frac{1}{2} \ln(2) \).

Key Concepts

Integration Techniques

Integration techniques are strategies that help evaluate integrals, especially when direct integration is complex. For the given problem, the integral \[ \int_{0}^{1/2} x \tan(\pi x^2) \, dx \]seems challenging due to the presence of the function \(x \tan(\pi x^2)\). A direct integration method might not work efficiently here, so we use a technique called **Substitution Method**, which simplifies the integral by transforming it into a more elementary form.Common integration techniques include:

• Substitution Method: Replaces variables to simplify the integration process.
• Integration by Parts: Splits the integral into parts that are easier to solve.
• Trigonometric Integration: Uses trigonometric identities to simplify and solve integrals.

By understanding these techniques, especially substitution used in our problem, we can tackle integrals that are initially daunting.

Substitution Method

The substitution method simplifies complex integrals by introducing a new variable. In our problem, we used the substitution \(u = \pi x^2\). This transforms our integral significantly. When performing substitution, follow these steps:

• Select a part of the integrand to substitute (here \(\pi x^2\)).
• Compute the differential for that substitution (\(du = 2\pi x \, dx\), so \(x \, dx = \frac{1}{2\pi} \, du\)).
• Adjust limits according to the new variable (from \(x = 0\) to \(x = \frac{1}{2}\), \(u\) changes from 0 to \(\frac{\pi}{4}\)).

By substituting, \[ \int_{0}^{1/2} x \tan(\pi x^2) \, dx \]becomes a simpler:\[ \int_{0}^{\pi/4} \frac{1}{2\pi} \tan(u) \, du \]This is often more straightforward to integrate.

Trigonometric Integration

Trigonometric integration involves using trigonometric identities to solve integrals. In the transformed integral:\[ \int_{0}^{\pi/4} \frac{1}{2\pi} \tan(u) \, du \]we need to integrate \( \tan(u) \).For \( \tan(u) \), we know the integral results in:\[-\ln |\cos(u)| + C\]where \(C\) is the constant of integration, not needed for definite integrals. This uses the identity:

• The derivative \(\frac{d}{du}[-\ln |\cos(u)|]\) gives \(\tan(u)\).

Trigonometric integrals often tap into such identities, greatly easing evaluation.

Antiderivatives

An antiderivative is the inverse operation of taking a derivative. In integration, the antiderivative of a function is necessary to evaluate the integral.For our problem, after substituting, integrating \(\tan(u)\) yielded:\[-\ln |\cos(u)|\]The antiderivative allows us to evaluate the definite integral by calculating:\[-\ln \left| \cos(\pi/4) \right| - (-\ln \left| \cos(0) \right|) \]Simplifying further takes advantage of known values in trigonometry, where:

• \(\cos(\pi/4) = \frac{1}{\sqrt{2}}\)
• \(\cos(0) = 1\)

Thus, the antiderivative assists in finding the integral's evaluation from the limits 0 to \(\frac{\pi}{4}\). Benefiting from known trigonometric identities during evaluation ensures a correct and simplified final answer.