Q.
Concept understanding — Integration by Decomposition and Partial Fractions
Some integrands have no direct standard formula but can be decomposed into a sum or difference of functions whose integrals are known, then integrated term by term. Typical decompositions: expanding a power such as , splitting a single fraction like into separate terms, and using trigonometric product-to-sum identities (e.g. , ), power-reduction (, ), and .
Partial fractions. When the integrand is a rational function with (if not, first do polynomial long division), resolve it into a sum of simpler fractions — one term per distinct linear factor (and for a repeated factor) — determine the constants by comparing coefficients or substituting convenient -values, then integrate each simple fraction (each gives a logarithm or a power).
If the numerator's degree is the denominator's, you must divide first; only the proper-fraction remainder is expanded into partial fractions.
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