se debe usar identidades trigonométicas
[tex]\cos(x-y)-\cos(x+y)=2\sin x\sin y\\ \\
\text{Por ende: }\\
\sin4x\sin3x = \dfrac{1}{2}\left(\cos x -\cos 7x\right)\\ \\
\text{As\'i:}\\ \\
\displaystyle
\int\sin4x\sin3x \,dx=\dfrac{1}{2}\int \cos x-\cos 7x \, dx\\ \\
\int\sin4x\sin3x \,dx=\dfrac{1}{2}\left(\sin x - \dfrac{1}{7}\sin7x\right)+C\\ \\ \\
\boxed{\int\sin4x\sin3x \,dx=\dfrac{1}{2}\sin x - \dfrac{1}{14}\sin7x+C}[/tex]
