Integration Decision Making

1. The 'Simple' Check

• First, determine if the function is a standard power of $x$, a simple polynomial, or a basic trigonometric function found in the formula booklet.
• If the expression contains brackets, consider if expanding them creates a simple polynomial that can be integrated term-by-term.

2. Fractional Integrands

• Algebraic Division: If the degree of the numerator is greater than or equal to the degree of the denominator (top-heavy), perform long division first.
• Logarithmic Form: Check if the numerator is a multiple of the derivative of the denominator, fitting the form $\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + c$.
• Partial Fractions: If the denominator is a polynomial that can be factored, decompose the fraction into simpler linear or quadratic components.

3. Product Integrands

• Reverse Chain Rule: If one part of the product is the derivative of the 'inner' part of a composite function, use the reverse chain rule or substitution.
• Integration by Parts: If the integrand is a product of two distinct types of functions (e.g., $x \sin(x)$), apply the formula $\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx$.

• It is vital to distinguish between when to use Substitution versus Integration by Parts. Substitution is for composite functions where the derivative of the inner function is present, while Parts is for products of unrelated functions.

• Another critical distinction is between Logarithmic Integration and Partial Fractions. Logarithmic integration is a single-step process for specific derivative ratios, whereas Partial Fractions is a pre-processing step to turn one complex fraction into several simpler ones.

• The 'Adjust and Compensate' Method: When using the reverse chain rule, always check if the derivative of the inner function matches the constant multiplier in the integrand. If it differs only by a constant, adjust the integral by multiplying by the reciprocal of that constant.

• Trigonometric Identities: Before integrating powers of $\sin(x)$ or $\cos(x)$, check if a double-angle identity (like $\cos(2x)$) can reduce the power to a linear trigonometric term.

• Verification: Always differentiate your final answer. If the result of your differentiation does not match the original integrand, an error occurred during the integration process.

• Sanity Check: In definite integration, ensure your limits are substituted into the correct variable, especially if you performed a substitution and did not change the limits to the new variable $u$.