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\( \quad \) \( \int \frac{8}{\sqrt{- 2 x - 1}} \, dx \)

\( =- 8 \sqrt{- 2 x - 1} + C \)

\( \quad \) \( \int^{\frac{- 5 \pi - 12}{12}}_{\frac{- 7 \pi - 12}{12}} - \frac{3}{\cos^{2}{\left(2 x + 2 \right)}} \, dx \)

\( = \left[ - \frac{3 \tan{\left(2 x + 2 \right)}}{2} \right]^{\frac{- 5 \pi - 12}{12}}_{\frac{- 7 \pi - 12}{12}} \)
\( = \left( - \frac{\sqrt{3}}{2} \right) - \frac{\sqrt{3}}{2} \)
\( = - \sqrt{3} \)

\( \quad \) \( \int - \frac{2 e^{2 x + 2}}{5} \, dx \)

\( =- \frac{e^{2 x + 2}}{5} + C \)

\( \quad \) \( \int^{-3}_{-5} - 2 \sqrt{- 2 x - 1} \, dx \)

\( = \left[ \left(- \frac{4 x}{3} - \frac{2}{3}\right) \sqrt{- 2 x - 1} \right]^{-3}_{-5} \)
\( = \frac{10 \sqrt{5}}{3} - 18 \)
\( = -18 + \frac{10 \sqrt{5}}{3} \)

\( \quad \) \( \int^{0}_{-1} - \frac{3 \cdot 4^{- x + 2}}{2} \, dx \)

\( = \left[ \frac{3\cdot4^{-x+2}}{2\log{4}} \right]^{0}_{-1} \)
\( = \frac{24}{\log{\left(4 \right)}} - \frac{96}{\log{\left(4 \right)}} \)
\( = - \frac{72}{\log{\left(4 \right)}} \)

\( \quad \) \( \int \frac{8}{\cos^{2}{\left(2 x - 2 \right)}} \, dx \)

\( =4 \tan{\left(2 x - 2 \right)} + C \)

\( \quad \) \( \int^{\frac{- 5 \pi + 3}{3}}_{- \pi + 1} \frac{\cos{\left(x - 1 \right)}}{2} \, dx \)

\( = \left[ \frac{\sin{\left(x - 1 \right)}}{2} \right]^{\frac{- 5 \pi + 3}{3}}_{- \pi + 1} \)
\( = \frac{\sqrt{3}}{4} - 0 \)
\( = \frac{\sqrt{3}}{4} \)

\( \quad \) \( \int^{\frac{- 7 \pi + 6}{6}}_{\frac{- 2 \pi + 3}{3}} \frac{12 \sin{\left(2 x - 2 \right)}}{5} \, dx \)

\( = \left[ - \frac{6 \cos{\left(2 x - 2 \right)}}{5} \right]^{\frac{- 7 \pi + 6}{6}}_{\frac{- 2 \pi + 3}{3}} \)
\( = \left( - \frac{3}{5} \right) - \frac{3}{5} \)
\( = - \frac{6}{5} \)

\( \quad \) \( \int^{3}_{1} - 6 \left(- x + 1\right)^{2} \, dx \)

\( = \left[ 2 \left(- x + 1\right)^{3} \right]^{3}_{1} \)
\( = \left( -16 \right) - 0 \)
\( = -16 \)

\( \quad \) \( \int 2 \sqrt{2 x - 2} \, dx \)

\( =\left(\frac{4 x}{3} - \frac{4}{3}\right) \sqrt{2 x - 2} + C \)