Can we integrate a function without using the ILATE rule? For instance, $$\int (2x+9x^2)\sin 4x \cos x\,dx.$$ Can I integrate it without using ILATE in a quicker time? Because when ILATE rule is followed, it gets into a loop.
2026-04-06 01:17:05.1775438225
Integration of a function.
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First, I think you need to simplify the $\sin(4x)\cos(x)$ using the following:
$$\sin(a)\cos(b)=\frac 12\left(\sin(a+b)+\sin(a-b)\right)$$ $$\sin(4x)\cos(x)=\frac 12 \left(\sin(5x)+\sin(3x)\right)$$
Expanding $\frac 12(2x+9x^2) \left(\sin(5x)+\sin(3x)\right)$ makes applying the ILATE rule (using integration by parts tables) pretty straightforward.