The first step is to identify the type of integral. Is it a standard integral, a simple substitution, integration by parts, or partial fractions? Recognizing the form will guide your approach.
Use the LIATE rule as a guide: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. u is usually the function that comes earlier in the LIATE order, as differentiating it often simplifies the integral.
Consider more advanced techniques like integration by parts or partial fractions. Integration by parts is useful for products of functions, while partial fractions is suitable for rational functions.
Differentiate your answer. If the derivative matches the original integrand, your integration is likely correct.
Use trigonometric substitution when the integral involves expressions of the form √(a² - x²), √(a² + x²), or √(x² - a²). Choose the appropriate trigonometric function to simplify the expression under the square root.
Always add + C to indefinite integrals. Remember that the derivative of a constant is zero, so there are infinitely many possible antiderivatives, differing only by a constant.