Practice section 4 of 10

Find (a) $\int\cos^2x\,dx$; (b) $\int2\sin^2(3\theta)\,d\theta$.

Using $\cos2A=2\cos^2A-1$, show that $\cos^4x=\frac38+\frac12\cos2x+\frac18\cos4x$. Hence find $\int\cos^4x\,dx$.

Find $\int\sin3x\cos3x\,dx$.

Given $2\cos A\cos B=\cos(A+B)+\cos(A-B)$ and $2\cos A\sin B=\sin(A+B)-\sin(A-B)$, find (a) $\int\cos3x\cos x\,dx$; (b) $\int\cos4x\sin x\,dx$.

Find (a) $\int x(2-3x)^{11}\,dx$, using $u=2-3x$; (b) $\int\frac{x+1}{x\sqrt{x-2}}\,dx$, using $u=\sqrt{x-2}$; (c) $\int\frac1{(x^2+4)^{3/2}}\,dx$, using $x=2\tan\theta$, $-\pi/2<\theta<\pi/2$; (d) $\int\frac{\sqrt{9-x^2}}{x^2}\,dx$, using $x=3\sin\theta$, $-\pi/2\le\theta\le\pi/2$.

Using $x=\sin\theta$, find $\int\sqrt{1-x^2}\,dx$. Compare the source extension using $x=\cos\theta$.