Integration by recognition is a powerful technique where we identify the integrand as the derivative of a known function. Instead of using complex integration methods, we recognize patterns from differentiation. For example, when we see x times cosine of x squared, we can recognize this as related to the derivative of sine of x squared.
The process follows five clear steps. First, identify the pattern in the integrand. Second, guess a related function whose derivative might match. Third, differentiate your guess using the chain rule. Fourth, adjust any constant factors if needed. Finally, add the constant of integration. Let's see this with an example involving exponential functions.
Recognizing common patterns is crucial for success. Look for the derivative of the inner function multiplied by a function of that inner function. Common patterns include polynomial times exponential, trigonometric derivatives, and logarithmic forms. Practice identifying these patterns will make integration by recognition much faster and more intuitive.
When the derivative doesn't match exactly, we need to adjust for constant factors. Compare the coefficients in your derivative with the original integrand. If the derivative is k times the integrand, then the antiderivative is one over k times your guessed function. In this example, the derivative matches perfectly, so no adjustment is needed.
Success with integration by recognition comes from practice and building pattern recognition skills. Start with simple examples and gradually work up to more complex forms. Always verify your answer by differentiating back to the original integrand. Build a mental library of common derivative patterns, and remember to always add the constant of integration. With practice, you'll quickly recognize when this method applies.