91Ó°ÊÓ

When integrating terms like \( x^2 \), the power rule is the key to simplifying these expressions. The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. This rule is straightforward and highly useful for polynomial functions.
It's essential to remember:

• To increase the power of \( x \) by one.
• To divide by the new power.

As seen, applying the power rule to \( x^2 \) gives us \( \frac{x^3}{3} \). This rule makes integrating polynomial terms efficiently without much hassle. Always ensure you adjust your powers correctly, and watch out for constants of integration.
For exercise problems like the one here, spotting terms that clearly fit the pattern of \( x^n \) will make your task easier when applying this rule. Consider this your go-to strategy for turning exponents into integrated expressions with ease.

Trigonometric functions like \( \cos 2x \) require a good understanding of their integral forms. When dealing with trigonometric integrals, it’s crucial to recognize how the function transforms. For a term like \( \cos kx \), the integral is \( \frac{1}{k} \sin kx + C \). This formula helps to integrate cosine functions effectively, ensuring that you adjust based on the coefficient in front of \( x \).
To successfully integrate trigonometric functions, remember:

• Simplify the expression by recognizing the "k" value, in this instance, \( k = 2 \).
• Adjust your integral accordingly, yielding \( \frac{1}{2} \sin 2x \).

Integrals of trigonometric functions provide a method to handle oscillating terms, converting them to manageable forms incorporated into the solution.
Understand that for trigonometric integration, precision in handling these constants right from the start will pave the way for precise solutions. The expression \( \frac{1}{2} \sin 2x \) results from these adjustments, facilitating accurate and efficient problem-solving.

Logarithmic integration usually comes into play when integrating functions like \( -\frac{1}{x} \), which can be rewritten as \( -x^{-1} \). The integral of \( x^{-1} \), or \( \frac{1}{x} \), is unique because it results in a natural logarithm: \( \ln |x| \). This method is vital for understanding when and how logarithms emerge from integration processes.
Key points to remember include:

• Identify terms which behave like \( \frac{1}{x} \).
• Remember, this leads to a natural logarithm, specifically \( -\ln|x| \) for this exercise due to the negative sign.

Integrating functions close to \( \frac{1}{x} \) returns logarithmic results, fundamentally different from polynomial integrals that yield higher-order powers. Always watch out for these terms as they add a different solution layer to the integral. For this reason, such terms should be carefully managed to produce a concise and correct expression for integrated functions.
Understanding the logarithmic integration identity helps in enhancing skills for seamlessly shifting from algebraic expressions to their logarithmic counterparts in calculus.