Problem 6 - Calculus Single Variable

Question

Integrate by parts to evaluate the given indefinite integral. $$ \int 9 x \cos (3 x) d x $$

Step-by-Step Solution

Answer

\(\int 9x \cos(3x) \, dx = 3x \sin(3x) + \cos(3x) + C \)

Step 1: Identify Parts

For integration by parts, we use the formula \( \int u \, dv = uv - \int v \, du \). Identify \( u \) and \( dv \) from the integral. In this case, set \( u = 9x \) and \( dv = \cos(3x) \, dx \).

Step 2: Differentiate and Integrate

Differentiate \( u = 9x \) to get \( du = 9 \, dx \). Integrate \( dv = \cos(3x) \, dx \) to obtain \( v = \frac{1}{3} \sin(3x) \).

Step 3: Substitute into Formula

Substitute \( u, v, \text{and} \, du \, \text{into the integration by parts formula}. \)\[\int 9x \cos(3x) \, dx = uv - \int v \, du = 9x \cdot \frac{1}{3} \sin(3x) - \int \frac{1}{3} \sin(3x) \cdot 9 \, dx\]

Step 4: Simplify the Expression

Simplify the expression:\[3x \sin(3x) - 3 \int \sin(3x) \, dx\]Now, integrate \( \int \sin(3x) \, dx = -\frac{1}{3} \cos(3x) \).

Step 5: Final Calculation

Substitute the integral from Step 4 back into the expression:\[3x \sin(3x) - 3 \left(-\frac{1}{3} \cos(3x)\right)\] Simplify to get:\[3x \sin(3x) + \cos(3x)\] Add the constant of integration, \( C \).

Step 6: Conclusion

Thus, the integral evaluates to:\[\int 9x \cos(3x) \, dx = 3x \sin(3x) + \cos(3x) + C\]

Key Concepts

Indefinite Integral

An indefinite integral represents the family of all antiderivatives of a function, which means finding a function whose derivative gives the original function. Unlike a definite integral, which calculates a specific numerical value over an interval, the indefinite integral results in a general formula that includes an arbitrary constant, known as the constant of integration, represented by \( C \). This constant appears because differentiation of a constant is zero, and thus when we integrate back, any constant could have been part of the original function.

The notation for indefinite integrals is \( \int f(x) \, dx \), read as "the integral of \( f(x) \) with respect to \( x \)." To solve an indefinite integral, one approaches it using various methods, such as basic antiderivatives, substitution, or more advanced techniques like integration by parts, which is used in this exercise.

Remember that indefinite integrals have the power to reverse the operation of differentiation, providing a way to recover a function from its rate of change.

Differentiation

Differentiation is the process of finding the derivative of a function, which indicates how a function changes at any point, essentially measuring the slope of the function. In calculus, differentiation helps to find rates of change, tangents to curves, and more.

In this exercise, differentiation is used to find \( du \) when applying the integration by parts formula. Specifically, we differentiated \( u = 9x \) to obtain \( du = 9 \, dx \). This is a straightforward differentiation where the derivative of \( x^n \), in this case \( x^1 \), is \( nx^{n-1} \). Here, the power of \( x \) is reduced by one, and the coefficient is factored out.

Differentiation serves as an essential step in many integration techniques, especially under integration by parts, where differentiating the chosen \( u \) helps simplify the integrand into a more manageable form.

Trigonometric Integration

Trigonometric integration involves integrating functions that include trigonometric functions like sine, cosine, and tangent. These types of integrals often require specific techniques or substitutions to be solved efficiently.

In our exercise, the trigonometric function \( \cos(3x) \) appears in the integral. To handle this, we first needed to integrate \( dv = \cos(3x) \, dx \) to get \( v \). The integration of a cosine function typically results in a sine function, as the derivative of sine is cosine. Therefore, \( \int \cos(3x) \, dx \) results in \( v = \frac{1}{3} \sin(3x) \) by using an internal substitution because the integral of \( \cos(ax) \) is \( \frac{1}{a} \sin(ax) \).

Trigonometric integrals often require carefully managing coefficients and pay close attention to substitutions to ensure the trigonometric identities are properly applied and solved, as shown when integrating other parts of the expression after applying integration by parts.