Problem 55 - Calculus Single Variable

Question

Evaluate the given integral by converting the integrand to an expression in sines and cosines. $$ \int 6 \cos ^{3}(x) \tan ^{2}(x) d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(6 \sin(x) - \frac{3}{2} \cos^4(x) + C\).

Step 1: Rewrite in Terms of Sine and Cosine

Start by expressing both \(\tan^2(x)\) and \(\cos^3(x)\) in terms of sine and cosine. Remember that \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) so \(\tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)}\). Therefore, we can rewrite the integral as: \[\int 6 \cos^3(x) \cdot \frac{\sin^2(x)}{\cos^2(x)} \, dx = \int 6 \cos(x) \sin^2(x) \, dx.\]

Step 2: Use a Trigonometric Identity

Next, use the identity \(\sin^2(x) = 1 - \cos^2(x)\) to simplify the integrand. Substitute to get:\[\int 6 \cos(x) (1 - \cos^2(x)) \, dx.\]

Step 3: Distribute and Simplify

Distribute \(\cos(x)\) over the term \((1 - \cos^2(x))\):\[\int 6 \cos(x) \, dx - \int 6 \cos^3(x) \, dx.\]

Step 4: Integrate Each Term

The first integral, \(\int 6 \cos(x) \, dx\), is straightforward:\[\int 6 \cos(x) \, dx = 6 \sin(x) + C_1.\]The second integral, \(\int 6 \cos^3(x) \, dx\), requires substitution. Let \(u = \cos(x)\), then \(du = -\sin(x) dx\). This changes the integral to:\[-6 \int u^3 \, du = -\frac{6}{4} u^4 + C_2 = -\frac{3}{2} \cos^4(x) + C_2.\]

Step 5: Combine Results

Combine both results from the integrals:\[6 \sin(x) - \frac{3}{2} \cos^4(x) + C,\]where \(C = C_1 + C_2\) is the constant of integration.

Key Concepts

Integration Techniques

Integration techniques are tools that help solve various calculus problems involving integrals. In this exercise, the strategy starts with recognizing how the integral can be simplified by using other mathematical expressions, such as converting trigonometric functions into more manageable forms.

• Identifying useful substitutions or identities allows for converting the integrand to expressions which are more straightforward to integrate.
• In our exercise, rewriting the integral in terms of sine and cosine functions simplifies the complexity.

In some cases, direct integration might not work, therefore, using these techniques is crucial. Learning when and how to apply specific strategies is key to mastering calculus integral problems.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for any angle. These are especially useful in calculus to simplify integrals involving trigonometric expressions.One of the most famed identities is the Pythagorean identity which relates sine and cosine:

• \( \sin^2(x) + \cos^2(x) = 1 \)
• Using this, the equation \( \sin^2(x) = 1 - \cos^2(x) \) helps simplify more complex integrals.

In the original solution, the identity \( \tan^2(x) = \frac{\sin^2(x)}{\cos^2(x)} \) and the Pythagorean identity were key in rewriting and simplifying the integral. This approach often makes the integral much simpler to solve.

Substitution Method

The substitution method is a powerful integration technique used to simplify integrals by changing the variable of integration. This often turns a complicated integral into a simpler one.

• To use substitution, a new variable \( u \) is defined, which is a function of the original variable \( x \).
• The differential \( dx \) is also transformed using the derivative of the substitution.

In the exercise, for the second integral, we use the substitution \( u = \cos(x) \). As a result, \( du = -\sin(x) dx \) is also determined, enabling the conversion of the entire integral into the variable \( u \). Substitution is particularly valuable when you encounter products of functions where one is easily differentiable.

Calculus Problem Solving

Solving calculus problems requires a systematic approach that often includes simplifying expressions, recognizing patterns, and applying appropriate techniques. It's not only about reaching an answer but understanding the process.

• Start by analyzing the given problem and identifying known forms or patterns.
• Apply relevant identities and transform the integrand into a simpler form before attempting integration.
• If needed, employ integration techniques such as substitution or partial fraction decomposition.

Once the integral is simplified and solved, don't forget to include the constant of integration, \( C \). This constant is crucial as it accounts for any additional constants that may arise when reversing the differentiation process. Understanding the logical flow between each step maximizes the understanding and success in solving calculus problems.