Kay's Math: Trig identities and integration

a. Prove that

$(2sin x+cos x)^{2}\equiv\frac{5}{2}+2 sin 2x-\frac{3}{2}cos2x$

b. Hence or otherwise

$\int_{0}^{\frac{\pi}{4}}(2sin x+cos x)^{2}dx$

a. Prove that

$12sin^{2}x cos^{2}x\equiv\frac{3}{2}(1-cos 4x)$

b. Hence or otherwise evaluate

$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}12sin^{2}x cos^{2}x dx$

3. a. Derive an expression for cos3x in terms of cos x only.

b. Hence

$\int_{0}^{\frac{\pi}{6}}2cos^{3}x-cos x\;dx$

Kay's Math