Q.$\sin^2 5x$

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 $(3x-1)^2$, splitting a single fraction like $\dfrac{x^2-3x+1}{x}$ into separate terms, and using trigonometric product-to-sum identities (e.g. $2\cos A\sin B=\sin(A+B)-\sin(A-B)$, $\cos^3 x=\tfrac14(3\cos x+\cos 3x)$), power-reduction ($1+\cos 2x=2\cos^2 x$, $1-\cos 2x=2\sin^2 x$), and $\sin^2 x+\cos^2 x=1$. …