Question

$\int(1+\cos3y)^2dy$

Explanation:

Step1: Expand the squared term

$(1+\cos3y)^2 = 1 + 2\cos3y + \cos^23y$

Step2: Use power-reduction identity

$\cos^23y = \frac{1+\cos6y}{2}$

Step3: Substitute and simplify integrand

$1 + 2\cos3y + \frac{1+\cos6y}{2} = \frac{3}{2} + 2\cos3y + \frac{1}{2}\cos6y$

Step4: Integrate term-by-term

$\int \frac{3}{2}dy + \int 2\cos3y dy + \int \frac{1}{2}\cos6y dy$
$\int \frac{3}{2}dy = \frac{3}{2}y$
$\int 2\cos3y dy = 2\cdot\frac{\sin3y}{3} = \frac{2}{3}\sin3y$
$\int \frac{1}{2}\cos6y dy = \frac{1}{2}\cdot\frac{\sin6y}{6} = \frac{1}{12}\sin6y$

Step5: Add constant of integration

Combine results and include $+C$

Answer:

$\frac{3}{2}y + \frac{2}{3}\sin3y + \frac{1}{12}\sin6y + C$