Problem 62 - Applied Calculus

Question

Find the indefinite integrals. $$ \int(12 \sin (2 x)+15 \cos (5 x)) d x $$

Step-by-Step Solution

Answer

\(-6 \cos(2x) + 3 \sin(5x) + C\)

Step 1: Understand the Term-wise Integration

The integral provided is composed of two separate terms: \(12 \sin(2x)\) and \(15 \cos(5x)\). To solve this integral, we can integrate each term separately and then sum the results.

Step 2: Integrate \(12 \sin(2x)\)

The integral of \(\sin(ax)\) is \(-\frac{1}{a} \cos(ax)\). Applying this formula, the integral of \(12 \sin(2x)\) becomes:\[12 \int \sin(2x) \, dx = 12 \left(-\frac{1}{2} \cos(2x) \right) = -6 \cos(2x)\]

Step 3: Integrate \(15 \cos(5x)\)

The integral of \(\cos(ax)\) is \(\frac{1}{a} \sin(ax)\). Applying this formula, the integral of \(15 \cos(5x)\) becomes:\[15 \int \cos(5x) \, dx = 15 \left(\frac{1}{5} \sin(5x) \right) = 3 \sin(5x)\]

Step 4: Combine the Results

Now that each term is integrated separately, combine the results to get the overall integral:\[-6 \cos(2x) + 3 \sin(5x) + C\]where \(C\) is the constant of integration.

Key Concepts

Integration Techniques

When tackling indefinite integrals, it's crucial to recognize and apply various integration techniques. These techniques simplify the task of finding the antiderivative of complex functions. Some common methods include:

• Basic Power Rule: Used mostly for polynomials, where you add one to the exponent and divide by the new exponent.
• Reverse Chain Rule: Handy for functions composed of an inner and outer function, particularly useful for trigonometric integrals.
• Integration by Substitution: Simplifies complicated functions by substituting a part of the equation with a single variable to make the equation easier to integrate.

In the given problem, the integral \(12 \sin(2x)+15 \cos(5x)\), we can utilize knowledge of trigonometric function integration to find the solution. By handling each term separately, and applying specific formulas for sine and cosine integrals, we employ a direct and efficient method. Focusing on individual terms allows us to break the problem into manageable parts which can be solved independently and later combined.

Trigonometric Integrals

Integrals involving trigonometric functions often require some specific strategies. Trigonometric functions such as \(\sin(x)\), \(\cos(x)\), \(\tan(x)\), and their multiples commonly appear in calculus problems. Fortunately, each has a specific antiderivative that can be directly applied:

• The integral of \(\sin(ax)\) is \(-\frac{1}{a}\cos(ax)\).
• The integral of \(\cos(ax)\) is \(\frac{1}{a}\sin(ax)\).

The exercise given includes \(12 \sin(2x)\) and \(15 \cos(5x)\). To integrate these terms, apply the respective formula:

• For \(12 \sin(2x)\), use the formula for \(\sin(ax)\), resulting in \(-6 \cos(2x)\).
• For \(15 \cos(5x)\), use the formula for \(\cos(ax)\), resulting in \(3 \sin(5x)\).

This approach focuses on recognizing patterns and matching them to known integral forms, simplifying the problem-solving process substantially.

Calculus Problem Solving

Effective calculus problem solving involves understanding and implementing solid strategies that streamline solution finding. Initially, break down problems into simpler parts, if possible. For instance, the integral \(\int(12 \sin (2 x)+15 \cos (5 x)) dx\) comprises two separate terms. Solving them individually makes comprehension and integration easier.
Always remember to:

• Identify distinct parts of complex problems that can be solved independently.
• Use relevant formulas for each part. In trigonometric integrals, direct formulas for sine and cosine functions were used.
• Don't forget to add the constant of integration \(C\), which accounts for any constant term that could have been present in the original function.

Thus, breaking down the problem, applying appropriate integration techniques, and then summing the results, leads to finding the complete solution. Mastering these elements equips students to handle tougher problems with confidence.