Question

How would I integrate this function?cos(x)[sqrt(1+cos^2(x))]

Answer 1

Use the fact that \(\cos^2x=1-\sin^2x\), so \(1+\cos^2x=2-\sin^2x\).
\[\int\cos x\sqrt{2-\sin^2x}\,dx\]
Set \(u=\sin x\), giving \(du=\cos x\,dx\), and so
\[\int\cos x\sqrt{2-\sin^2x}\,dx=\sqrt{2-u^2}\,du\]
Make another substitution, \(u=\sqrt2\sin t\), so \(du=\sqrt2\cos t\,dt\).
\[\begin{align*}\sqrt{2-u^2}\,du&=\sqrt2\int\sqrt{2-(\sqrt2\sin t)^2}\cos t\,dt\\\\&=2\int\sqrt{1-\sin^2t}\,dt\\\\&=2\int\cos t\,dt\end{align*}\]

Answer 2

Ahhh, Thank you!!!