2023 MI P1 Q10

Differentiate \({{\sin }^{-1}}\left( {{\mathrm{e}}^{2x}} \right)\) with respect to \(x\).[2]

Hence find \(\displaystyle\displaystyle \displaystyle\int{\frac{2{{\mathrm{e}}^{2x}}+{{\mathrm{e}}^{4x}}}{\sqrt{1-{{\mathrm{e}}^{4x}}}}}\,\mathrm{d}x\).[3]

Show that \({{\sin }^{n-2}}\theta {{\cos }^{2}}\theta ={{\sin }^{n-2}}\theta -{{\sin }^{n}}\theta \), where \(n\in {{\mathbb{Z}}^{+}}\), \(n\ge 2\).[1]

By considering \({{\sin }^{n}}\theta ={{\sin }^{n-1}}\theta \sin \theta \), show that

\(\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi }{2}}{{{\sin }^{n}}\theta \,\,\mathrm{d}\theta =\frac{n-1}{n}\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi }{2}}{{{\sin }^{n-2}}\theta \,\,\mathrm{d}\theta }}\),

where \(n\in {{\mathbb{Z}}^{+}}\), \(n\ge 2\).[4]

Hence find the exact value of \(\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi }{2}}{{{\sin }^{3}}\theta \,\,\mathrm{d}\theta }\).[2]