Decomposition method

Not every integrand matches one of the standard formulas directly. The decomposition method handles this by splitting a single "hard" integrand into a sum or difference of two or more "easy" pieces, each of which integrates by a known formula, and then integrating termwise using linearity: $\int[f(x)\pm g(x)]dx = \int f(x)dx \pm \int g(x)dx$.

The functions that call for this trick are typically algebraic, trigonometric, or exponential expressions that don't have their own listed formula — e.g. $(1-x^3)^2$, $\dfrac{x^2-x+1}{x^3}$, $\cos5x\sin3x$, $\cos^3x$, $\dfrac{e^{2x}-1}{e^x}$ — but which decompose into pieces that do.

Algebraic decomposition. A power of a binomial like $(1-x^3)^2$ is expanded first ($=1-2x^3+x^6$) and then integrated term by term. A fraction whose numerator has several terms over a monomial denominator, such as $\dfrac{x^2-x+1}{x^3}$, is split term-by-term into $\dfrac1x-\dfrac1{x^2}+\dfrac1{x^3}$ and each piece integrated using the power rule (with $\int x^{-1}dx=\log|x|+c$ for the middle term). A quotient like $\dfrac{(x-1)^2}{x^3+x}$ is handled by expanding the numerator and splitting the resulting fraction over the factored denominator $x(x^2+1)$ into $\dfrac1x-\dfrac{2x}{x^2+1}$, each piece a known log-type integral — this is really a first taste of the partial-fractions idea developed fully in §11.7.2.

Trigonometric decomposition. A product of two different trig functions of different arguments, like $\cos5x\sin3x$, is converted to a sum using a product-to-sum identity (e.g. $2\cos A\sin B=\sin(A+B)-\sin(A-B)$) before integrating. A power like $\cos^3x$ is reduced using the triple-angle identity $\cos^3x=\dfrac14(3\cos x+\cos3x)$. An expression like $\dfrac{1}{\sin^2x\cos^2x}$ is attacked by writing $1=\sin^2x+\cos^2x$ in the numerator and splitting into $\sec^2x+\csc^2x$. A quotient such as $\dfrac{\sin x}{1+\sin x}$ is rationalised by multiplying top and bottom by $1-\sin x$, turning the denominator into $1-\sin^2x=\cos^2x$ and the whole thing into $\tan x\sec x-\sec^2x+1$, three standard pieces. Expressions built from $1\pm\cos2x$ or $1\pm\sin2x$ are simplified first with the half-angle/double-angle identities $1+\cos2x=2\cos^2x$, $1-\cos2x=2\sin^2x$, and $1\pm\sin2x=(\cos x\pm\sin x)^2$, which often collapses a square-root integrand into something linear in $\sin x,\cos x$. A square like $(\tan x+\cot x)^2$ expands to $\tan^2x+2+\cot^2x$ and then converts to $\sec^2x+\csc^2x$ via the Pythagorean identities $\tan^2x=\sec^2x-1$, $\cot^2x=\csc^2x-1$.

Exponential decomposition. A quotient like $\dfrac{e^{2x}-1}{e^x}$ is split by dividing each term of the numerator by $e^x$, giving $e^x-e^{-x}$. A product of two different exponential bases, such as $a^xe^x$, is recognised as a single exponential with base $ae$: $a^xe^x=(ae)^x$, integrating to $\dfrac{(ae)^x}{\log(ae)}+c$ (from $\int b^xdx=\dfrac{b^x}{\log b}+c$). …