Answer
$$-\frac{1}{7}\sin^7x +\frac{3}{5}\sin^5x-\sin^3x+\sin x+C$$
Work Step by Step
\begin{align*}
\int\cos^7xdx&=\int\cos^6x\cos xdx\\
&=\int (1-\sin^2 x)^3\cos xdx\\
&=\int (-\sin^6x+3\sin^4x-3\sin^2x+1)\cos xdx\\
&=-\frac{1}{7}\sin^7x +\frac{3}{5}\sin^5x-\sin^3x+\sin x+C
\end{align*}