Alternate solutions to similar integrals? So, when I started this problem I couldn't see any reasonable trig substitutions that would make it simpler, and I don't know of a better method, so I attempted to solve it using repeated use of integration by parts. Is there a much easier, better, alternate solution to integrals of this nature in general? I'd also like to know where I went wrong with this solution.
$$
\begin{align*}
\int\sin^4x\,\mathrm{d}x&\rightarrow\\
u_1&=\sin^4x\\
u_1'&=4\cos x\sin^3x\\
v_1&=x\\
v_1'&=1\\
\leftarrow\int\sin^4x\,\mathrm{d}x&=x\sin^4x-4\int x\cos x\sin^3x\,\mathrm{d}x\\
\rightarrow\int x\cos x\sin^3x\,\mathrm{d}x&\rightarrow\\
u_2&=\sin^3x\\
u_2'&=3\cos x\sin^2x\\
v_2&=\int x\cos x\,\mathrm{d}x\\
v_2'&=x\cos x\\
\rightarrow\int x\cos x\,\mathrm{d}x\rightarrow\\
u_3&=x\\
u_3'&=1\\
v_3&=\sin x\\
v_3'&=\cos x\\
\leftarrow\int x\cos x\,\mathrm{d}x&=x\sin x-\int\sin x\,\mathrm{d}x\\
&=x\sin x+\cos x\\
\leftarrow\int x\cos x\sin^3x\,\mathrm{d}x&=\left(\sin^3x\right)\left(x\sin x+\cos x\right)-\int\left(3\cos x\sin^2x\right)\left(x\sin x+\cos x\right)\,\mathrm{d}x\\
&=x\sin^4x+\cos x\sin^3x-3\int x\cos x\sin^3 x\,\mathrm{d}x+3\int\cos^2
x\sin^2x\,\mathrm{d}x\\
4\int x\cos x\sin^3x\,\mathrm{d}x&=x\sin^4
x+\cos x\sin^3 x+3\int\cos^2x\sin^2x\,\mathrm{d}x\\
&=x\sin^4
x+\cos x\sin^3 x+3\int\left(1-\sin^2x\right)\sin^2x\,\mathrm{d}x\\
&=x\sin^4
x+\cos x\sin^3 x+3\int\sin^2x\,\mathrm{d}x-\int\sin^4x\,\mathrm{d}x\\
\rightarrow\int\sin^2x\,\mathrm{d}x\rightarrow\\
\cos2x&=1-2\sin^2x\\
\sin^2x&=\frac{1}{2}(1-\cos2x)\\
\leftarrow\frac{1}{2}\int1-\cos2x\,\mathrm{d}x&=\frac{1}{2}x-\frac{1}{4}\sin{2x}\\
\leftarrow4\int x\cos x\sin^3x\,\mathrm{d}x&=x\sin^4
x+\cos x\sin^3 x+3\left[\frac{1}{2}x-\frac{1}{4}\sin{2x}\right]-\int\sin^4x\,\mathrm{d}x\\
&=x\sin^4 x+\cos x\sin^3 x+\frac{3}{2}x-\frac{3}{4}\sin{2x}-\int\sin^4x\,\mathrm{d}x\\
\leftarrow\int\sin^4x\,\mathrm{d}x&=x\sin^4x-\left[x\sin^4 x+\cos x\sin^3 x+\frac{3}{2}x-\frac{3}{4}\sin{2x}-\int\sin^4x\,\mathrm{d}x\right]\\
&=x\sin^4x-x\sin^4x-\cos x\sin^3x-\frac{3}{2}x+\frac{3}{4}\sin2x+\int\sin^4x\,\mathrm{d}x\\
\end{align*}
$$
As you can see, I can't continue because the integral cancels itself.

Consider:$$\int\sin^4(x)\,\mathrm{d}x$$Apply the reduction formula:$$=\frac{3}{4}\cdot\int\sin^2(x)\,\mathrm{d}x\,-\frac{\cos(x)\sin^3(x)}{x}$$$$=\frac{3}{4}\cdot\left(\frac{1}{2}x-\frac{\cos(x)\sin(x)}{x}\right)-\frac{\cos(x)\sin^3(x)}{x}$$$$\boxed{=\frac{\sin(4x)-8\sin(2x)+12x}{32}+C}$$
