\displaystyle \textbf{Question 1: }~\int \sin^2(2x+5)\,dx.
\displaystyle \text{Answer:}

\displaystyle \int \sin^2(2x+5)\,dx

\displaystyle =\int \left(\frac{1-\cos(4x+10)}{2}\right)\,dx

\displaystyle =\frac{1}{2}\int \left(1-\cos(4x+10)\right)\,dx

\displaystyle =\frac{1}{2}\left[x-\frac{\sin(4x+10)}{4}\right]+C

\displaystyle =\frac{1}{2}x-\frac{\sin(4x+10)}{8}+C

\displaystyle \textbf{Question 2: }~\int \sin^3(2x+1)\,dx.
\displaystyle \text{Answer:}

\displaystyle \int \sin^3(2x+1)\,dx

\displaystyle =\frac{1}{4}\int \left[3\sin(2x+1)-\sin(6x+3)\right]\,dx

\displaystyle =\frac{3}{4}\int \sin(2x+1)\,dx-\frac{1}{4}\int \sin(6x+3)\,dx

\displaystyle =\frac{3}{4}\left[-\frac{\cos(2x+1)}{2}\right]-\frac{1}{4}\left[-\frac{\cos(6x+3)}{6}\right]+C

\displaystyle =-\frac{3}{8}\cos(2x+1)+\frac{1}{24}\cos(6x+3)+C

\displaystyle \textbf{Question 3: }~\int \cos^4 2x\,dx.
\displaystyle \text{Answer:}

\displaystyle \int \cos^4 2x\,dx

\displaystyle =\int (\cos^2 2x)^2\,dx

\displaystyle =\int \left(\frac{1+\cos 4x}{2}\right)^2 dx

\displaystyle =\frac{1}{4}\int (1+2\cos 4x+\cos^2 4x)\,dx

\displaystyle =\frac{1}{4}\int \left[1+2\cos 4x+\frac{1+\cos 8x}{2}\right]dx

\displaystyle =\frac{1}{4}\int \left(\frac{3}{2}+2\cos 4x+\frac{\cos 8x}{2}\right)dx

\displaystyle =\frac{1}{4}\left[\frac{3x}{2}+\frac{\sin 4x}{2}+\frac{\sin 8x}{16}\right]+C

\displaystyle =\frac{3x}{8}+\frac{\sin 4x}{8}+\frac{\sin 8x}{64}+C

\displaystyle \textbf{Question 4: }~\int \sin^2(bx)\,dx.
\displaystyle \text{Answer:}

\displaystyle \int \sin^2 bx\,dx

\displaystyle =\int \left(\frac{1-\cos 2bx}{2}\right)\,dx

\displaystyle =\frac{1}{2}\int \left(1-\cos 2bx\right)\,dx

\displaystyle =\frac{1}{2}\left[x-\frac{\sin 2bx}{2b}\right]+C

\displaystyle =\frac{x}{2}-\frac{\sin 2bx}{4b}+C

\displaystyle \textbf{Question 5: }~\int \sin^2\!\left(\frac{x}{2}\right)\,dx.
\displaystyle \text{Answer:}

\displaystyle \int \sin^2\frac{x}{2}\,dx

\displaystyle =\int \left(\frac{1-\cos x}{2}\right)\,dx

\displaystyle =\frac{1}{2}\int (1-\cos x)\,dx

\displaystyle =\frac{1}{2}\left[x-\sin x\right]+C

\displaystyle \textbf{Question 6: }~\int \cos^2\!\left(\frac{x}{2}\right)\,dx.
\displaystyle \text{Answer:}

\displaystyle \int \cos^2\frac{x}{2}\,dx

\displaystyle =\int \left(\frac{1+\cos x}{2}\right)\,dx

\displaystyle =\frac{1}{2}\int (1+\cos x)\,dx

\displaystyle =\frac{1}{2}\left[x+\sin x\right]+C

\displaystyle \textbf{Question 7: }~\int \cos^2(nx)\,dx.
\displaystyle \text{Answer:}

\displaystyle \int \cos^2 nx\,dx

\displaystyle =\int \left(\frac{1+\cos 2nx}{2}\right)\,dx

\displaystyle =\frac{1}{2}\int (1+\cos 2nx)\,dx

\displaystyle =\frac{1}{2}\left[x+\frac{\sin 2nx}{2n}\right]+C

\displaystyle =\frac{x}{2}+\frac{\sin 2nx}{4n}+C

\displaystyle \textbf{Question 8: }~\int \sin x\,\sqrt{1-\cos 2x}\,dx.
\displaystyle \text{Answer:}

\displaystyle \int \sin x\,\sqrt{1-\cos 2x}\,dx

\displaystyle =\int \sin x\,\sqrt{2\sin^2 x}\,dx

\displaystyle =\sqrt{2}\int \sin^2 x\,dx

\displaystyle =\sqrt{2}\int \left(\frac{1-\cos 2x}{2}\right)\,dx

\displaystyle =\frac{1}{\sqrt{2}}\int (1-\cos 2x)\,dx

\displaystyle =\frac{1}{\sqrt{2}}\left[x-\frac{\sin 2x}{2}\right]+C

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