Practice section 8 of 8

Show that $\sin^4(\theta/2)\equiv\dfrac18(3-4\cos\theta+\cos2\theta)$.

Express $3\sin x-4\cos x$ in the form $r\sin(x-\alpha)$. Show that $0\le3\sin x-4\cos x+5\le10$.

Find $\dfrac{dy}{dx}$ if $y=\sqrt{\dfrac3{2x+1}}$.

Differentiate $x^3\ln(2x+1)$ with respect to $x$.

Integrate $[(3x-1)^2+\cos(2x)]$ with respect to $x$.