Problem 11 - Calculus: One and Several Variables

Question

Calculate. (If you run out of ideas, use the examples as models.) $$\int \tan ^{3} x d x$$.

Step-by-Step Solution

Answer

The integral of \(\tan^3x\) is \( \tan{x} \cdot (\tan{x} - x) + \tan{x} + x \cdot \sec^2{x} + C\).

Step 1: Rewrite using trigonometric identity

Rewrite \(\tan^3x\) as \(\tan^2x \cdot \tan{x}\) and remember the trigonometric identity \(1 + \tan^2x = \sec^2x\). This gives us \(\int (\sec^2x - 1) \cdot \tan{x} dx\)

Step 2: Apply the method of Integration by Parts

Following the formula for integration by parts \(\int u \cdot v dx = u \cdot \int v dx - \int u’ (\int v dx) dx\), let \(u = \tan{x}\), which means \(du = \sec^2{x} dx\), and let \(dv = (\sec^2x - 1) dx\), which gives \(v = \tan{x} - x\). Applying the formula leads to: \( \tan{x} \cdot \int (\sec^2x - 1) dx - \int ( \tan{x} - x ) \cdot \sec^2{x} dx\). Now, we just need to calculate these two integrals separately.

Step 3: Simplify the integral

The first integral simplifies to \( \tan{x} \cdot (\tan{x} - x)\), and for the second integral, note that the integral of \(x \cdot \sec^2{x}\) is not easily solved, but fortunately its derivative gives \( \tan{x} + x \cdot \sec^2{x}\) which is the integrand we have, hence this entire expression is actually \(- \tan{x} - x \cdot \sec^2{x} + C\), where C is the constant of integration. We can now substitute these results back.

Step 4: Construct the final integral

Finally, replace back the calculated integrals to get the integral of \(\tan^3x\) as: \( \tan{x} \cdot (\tan{x} - x) + \tan{x} + x \cdot \sec^2{x} + C\).

Key Concepts

Trigonometric Integration

When working with trigonometric integration, we often deal with integrals that involve trigonometric functions like sine, cosine, or tangent. These integrations require smart techniques to simplify them. For the integral \( \int \tan^3 x \, dx \), we utilize trigonometric identities to make the problem more manageable.
A key trigonometric identity used here is \( \tan^2 x = \sec^2 x - 1 \). Recognizing such identities helps you rewrite complex expressions into simpler forms involving functions whose integrals are easier to compute. By splitting the expression \( \tan^3 x \) into \( \tan^2 x \cdot \tan x \), and substituting \( \tan^2 x \) with \( \sec^2 x - 1 \), we gain leverage over a seemingly tough integral by transforming it into a manageable state:

• \( \int (\sec^2 x - 1) \cdot \tan x \, dx \)

This form paves the way for applying more advanced calculus techniques, such as integration by parts, by leveraging the neat trick of breaking down trigonometric powers for integration.

Integral Calculus

Integral Calculus involves the computation of integrals and is essential for finding areas, volumes, and solving various types of differential equations. In our particular problem, we encounter the integral \( \int \tan^3 x \, dx \). Calculating this requires a structured method, one standout tool being Integration by Parts, a technique derived from the product rule of differentiation. This formula is expressed as:

• \( \int u \, dv = u \, v - \int v \, du \)

This approach is particularly useful when dealing with the product of two different types of functions, as is the case here with \( \tan x \cdot (\sec^2 x - 1) \). The challenge lies in choosing which part of the function will be \( u \) and which will be \( dv \).
By setting \( u = \tan x \) and \( dv = (\sec^2 x - 1) \, dx \), we can apply Integration by Parts effectively:

• \( du = \sec^2 x \, dx \)
• \( v = \tan x - x \)

Hence, the integration by parts helps break down complex problems into manageable steps, each step solving a part of the integral to eventually find a comprehensive solution.

Calculus Techniques

In calculus, leveraging different techniques is crucial for solving contrasting problems efficiently. Various methods like substitution, partial fractions, or Integration by Parts play important roles. Specifically, Integration by Parts stands out in our problem of integrating \( \int \tan^3 x \, dx \).

• First, identify parts of the integral that can be designated as \( u \) and \( dv \), making the function simpler to manipulate.
• Calculate derivatives and antiderivatives as needed to apply the technique.

In our problem, we successfully used Integration by Parts by selecting \( u = \tan x \) and \( dv = (\sec^2 x - 1) \, dx \), computing \( v \) and \( du \) as needed.
Once solved, simplifying the result (by recognizing derivative properties that revert us back to original expressions) is key to ensure you reach the simplest form. Continual practice with identifying proper techniques and substitutions builds proficiency in tackling various complicated integrals, streamlining the transition from problem to solution effectively.