Problem 47 - Calculus Single Variable

Question

Evaluate the given integral by converting the integrand to an expression in sines and cosines. $$ \int 3 \tan (x) \sec ^{3}(x) d x $$

Step-by-Step Solution

Answer

The evaluated integral is \( \sec^3(x) + C \).

Step 1: Recall the Trigonometric Identities

Recall the trigonometric identities \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) and \( \sec(x) = \frac{1}{\cos(x)} \). Use these identities to express both \( \tan(x) \) and \( \sec(x) \) in terms of \( \sin(x) \) and \( \cos(x) \).

Step 2: Substitute and Simplify the Integrand

Substitute the identities into the integral: \[ \int 3 \tan(x) \sec^3(x) dx = \int 3 \left(\frac{\sin(x)}{\cos(x)}\right) \left(\frac{1}{\cos(x)}\right)^3 dx = \int 3 \frac{\sin(x)}{\cos^4(x)} dx \] This simplifies the expression in terms of sine and cosine.

Step 3: Use Substitution to Simplify Integrals

Set \( u = \cos(x) \), so \( du = -\sin(x) \, dx \). Rewrite the integral using \( u \): \[ \int 3 \frac{\sin(x)}{\cos^4(x)} dx = \int 3 \frac{-du}{u^4} \] This substitution transforms the integral into a more standard form.

Step 4: Integrate Using Power Rule

Integrate the expression using the power rule. \[ \int -3 u^{-4} \, du = -3 \cdot \left(\frac{u^{-3}}{-3}\right) + C = u^{-3} + C \] This integral evaluates with the integration constant \( C \).

Step 5: Substitute Back in Terms of x

Replace \( u \) with \( \cos(x) \): \[ u^{-3} = \frac{1}{u^3} = \frac{1}{\cos^3(x)} = \sec^3(x) \] Thus, the solution is: \[ \sec^3(x) + C \] This final form represents the evaluated integral in terms of \( x \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in calculus for simplifying expressions and solving integrals. They allow us to express trigonometric functions like tangent and secant in terms of more fundamental functions such as sine and cosine. Understanding these identities is crucial for integral calculus.

• Tangent and Secant: The identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) expresses the tangent function as a ratio of sine to cosine.
• Secant as a Reciprocal: Another vital identity is \( \sec(x) = \frac{1}{\cos(x)} \), which shows the secant as the reciprocal of cosine.

By transforming tangent and secant in terms of sine and cosine, we can simplify complex integrals. This approach is particularly beneficial when tackling integrals involving products or powers of trigonometric functions. Simplifying expressions in this way lays the groundwork for further integration techniques.

Substitution Method

The substitution method is a fundamental technique used in solving integrals by simplifying complex expressions into more manageable forms. This method is particularly useful when dealing with integrals that resemble compositions of functions. It involves selecting a substitution, or change of variable, that reduces the integral into a simpler form.

Here's how it works:

• Choosing a Substitution: First, identify a part of the integrand that can be set as a new variable \( u \). This choice should simplify the integral immensely. In our example, substituting \( u = \cos(x) \) was strategic.
• Computing \( du \): Calculate the derivative of your substitution, \( du = -\sin(x) \, dx \), to replace the remaining part of the integrand.
• Rewriting the Integral: Substitute \( u \) and \( du \) into the integral, transforming it into a basic form that is easier to handle.
• Integrating: Perform the integration on the simpler expression, then revert back to the original variable.

By employing the substitution method, seemingly daunting integrals become much more straightforward, opening the path to solution.

Integration Techniques

Integration techniques are strategies used to solve integrals, focusing on transforming complicated expressions into simpler ones. Calculus students often leverage these techniques to evaluate integrals efficiently.

For example, the integral \( \int 3 \frac{-du}{u^4} \) looks complex, but with integration by the power rule, it becomes quite simple:

• Power Rule: For integrals of the form \( \int u^n \, du \), the formula is \( \frac{u^{n+1}}{n+1} \), provided \( n eq -1 \).
• Applying the Rule: Here, \( n = -4 \), so we integrate by adding 1 to the exponent: \( -3 \int u^{-4} \, du = -3 \left(-\frac{u^{-3}}{3}\right) \).
• Simplifying the Result: Simplifying gives \( u^{-3} \), which we then convert back to the original trigonometric expression: \( \sec^3(x) \).

Integration techniques involve not just applying formulas but also a significant understanding of when and how to apply them effectively. Mastering these strategies requires practice and familiarity with different types of functions and their behaviors.