Suppose we let [tex]u=\sin\theta+\cos\theta[/tex], so that [tex]\mathrm du=(\cos\theta-\sin\theta)\,\mathrm d\theta[/tex].
Also, recall the double angle identity for cosine:
[tex]\cos(2\theta)=\cos^2\theta-\sin^2\theta=(\cos\theta-\sin\theta)(\cos\theta+\sin\theta)[/tex]
So, we can rewrite and compute the integral using the substitution, as
[tex]\displaystyle\int\cos(2\theta)(\sin\theta+\cos\theta)^3\,\mathrm d\theta[/tex]
[tex]=\displaystyle\int u\cdot u^3\,\mathrm du[/tex]
[tex]=\displaystyle\int u^4\,\mathrm du[/tex]
[tex]=\dfrac{u^5}5+C[/tex]
[tex]=\boxed{\dfrac{(\cos\theta+\sin\theta)^5}5+C}[/tex]
