\ \ 1-2Sin^2\alpha=Cos2\alpha \ \ => \\
=> \ \ Sin^2\alpha=\frac{1+Cos2\alpha}{2} \\
\int {Sin^2x\cdot Cos^2x} \, dx =\frac{1}{8}\int {(1+Cos4x)dx=\frac{1}{8}( \int{}dx+\int{Cos4x}dx)" alt="Sin^2x\cdot Cos^2x=(Sinx\cdot Cosx)(Sinx\cdot Cosx)=\frac{1}{4}(Sin2x)(Sin2x)= \\
=\frac{1}{4}Sin^22x \\
Cos^2\alpha -Sin^2\alpha=Cos2\alpha \ \ => \ \ 1-2Sin^2\alpha=Cos2\alpha \ \ => \\
=> \ \ Sin^2\alpha=\frac{1+Cos2\alpha}{2} \\
\int {Sin^2x\cdot Cos^2x} \, dx =\frac{1}{8}\int {(1+Cos4x)dx=\frac{1}{8}( \int{}dx+\int{Cos4x}dx)" align="absmiddle" class="latex-formula">
\ \ 4xdx=du \ \ => \ \ \frac{du}{4}=dx \ \ => \\
=> \ \ \int{Cos4x}dx=\frac{1}{4}\int{Cosu}du=\frac{Sinu}{4}=\frac{Sin4x}{4} \\
\int {Cos4x} dx=\frac{Sin4x}{4}" alt="\int {}dx=x \\
\int {Cos4x} dx: \\
4x=u \ \ => \ \ 4xdx=du \ \ => \ \ \frac{du}{4}=dx \ \ => \\
=> \ \ \int{Cos4x}dx=\frac{1}{4}\int{Cosu}du=\frac{Sinu}{4}=\frac{Sin4x}{4} \\
\int {Cos4x} dx=\frac{Sin4x}{4}" align="absmiddle" class="latex-formula">
