■ No GDC ■(a) (i) \(\displaystyle\int_0^a {{x^2}{\rm{d}}x} = \left. {\frac{1}{3}{x^3}} \right]_0^a = \frac{1}{3}{a^3} - \frac{1}{3}{0^3} = \frac{1}{3}{a^3}\) Q.E.D. (ii) \(\displaystyle\frac{{\rm{d}}}{{{\rm{d}}a}}\left( {\frac{1}{3}{a^3}} \right) = {a^2}\)(b) (i) \(\displaystyle\int_0^a {\cos \left( x \right)\,{\rm{d}}x} = \left. {\sin \left( x \right)} \right]_0^a = \sin \left( a \right) - \sin \left( 0 \right) = \sin \left( a \right) - 0 = \sin \left( a \right)\) Q.E.D. (ii) \(\dfrac{{\rm{d}}}{{{\rm{d}}a}}\left( {\sin \left( a \right)} \right) = \cos \left( a \right)\)(c) (i) \(\displaystyle\int_0^a {\sqrt x \,{\rm{d}}x} = \int_0^a {{x^{\frac{1}{2}}}\,{\rm{d}}x} = \left. {\frac{2}{3}{x^{\frac{3}{2}}}} \right]_0^a = \frac{2}{3}{a^{\frac{3}{2}}} - \frac{2}{3}{0^{\frac{3}{2}}} = \frac{2}{3}\sqrt {{a^3}} \) Q.E.D. (ii) \(\displaystyle\frac{{\rm{d}}}{{{\rm{d}}a}}\left( {\frac{2}{3}\sqrt {{a^3}} } \right) = \frac{{\rm{d}}}{{{\rm{d}}a}}\left( {\frac{2}{3}{a^{\frac{3}{2}}}} \right) = {a^{\frac{1}{2}}} = \sqrt a \)(d) (i) \(\displaystyle\int_0^a {{{\rm{e}}^x}{\rm{d}}x} = \left. {{{\rm{e}}^x}} \right]_0^a = {{\rm{e}}^a} - {{\rm{e}}^0} = {{\rm{e}}^a} - 1\) Q.E.D. (ii) \(\dfrac{{\rm{d}}}{{{\rm{d}}a}}\left( {{{\rm{e}}^a} - 1} \right) = {{\rm{e}}^a} - 0 = {{\rm{e}}^a}\)(e) (i) \(\displaystyle\int_0^a {\sin \left( x \right)\,{\rm{d}}x} = \left. { - \cos \left( x \right)} \right]_0^a = - \cos \left( a \right) - \left( { - \cos \left( 0 \right)} \right) = - \cos \left( a \right) - \left( { - 1} \right) = 1 - \cos \left( a \right)\) Q.E.D. (ii) \(\dfrac{{\rm{d}}}{{{\rm{d}}a}}\left( {1 - \cos \left( a \right)} \right) = 0 - \left( { - \sin \left( a \right)} \right) = \sin \left( a \right)\)(f) (i) \(\displaystyle\int {u\,{\rm{d}}v} = uv - \int {v\,{\rm{d}}u} \) \(\displaystyle\int {\ln \left( x \right)\,{\rm{d}}x} \), let \(u = \ln \left( x \right)\) and \({\rm{d}}v = {\rm{d}}x\;\;\; \Rightarrow \;\;\;\)\({\rm{d}}u = \dfrac{1}{x}{\rm{d}}x\) and \(v = x\) then \(\displaystyle\int {\ln \left( x \right)\,{\rm{d}}x} = x\ln \left( x \right) - \int {x \cdot \frac{1}{x}{\rm{d}}x} = x\ln \left( x \right) - \int {{\rm{d}}x} = x\ln \left( x \right) - x\) \(\displaystyle\int_0^a {\ln \left( x \right)\,{\rm{d}}x} = \left. {x\ln \left( x \right) - x} \right]_0^a = \left( {a\ln \left( a \right) - a} \right) - \left( {0\ln \left( 0 \right) - 0} \right) = a\ln \left( a \right) - a\) Q.E.D. (ii) \(\displaystyle\frac{{\rm{d}}}{{{\rm{d}}a}}\left( {a\ln \left( a \right) - a} \right) = 1 \cdot \ln \left( a \right) + a \cdot \frac{1}{a} - 1 = \ln \left( a \right)\)