■ No GDC ■for HL students(a) (i) Show that \(\displaystyle\int_0^a {{x^2}{\rm{d}}x} = \frac{1}{3}{a^3}\) (ii) Find \(\displaystyle\frac{{\rm{d}}}{{{\rm{d}}a}}\left( {\frac{1}{3}{a^3}} \right)\)(b) (i) Show that \(\displaystyle\int_0^a {\cos \left( x \right)\,{\rm{d}}x} = \sin \left( a \right)\) (ii) Find \(\dfrac{{\rm{d}}}{{{\rm{d}}a}}\left( {\sin \left( a \right)} \right)\)(c) (i) Show that \(\displaystyle\int_0^a {\sqrt x \,{\rm{d}}x} = \frac{2}{3}\sqrt {{a^3}} \) (ii) Find \(\dfrac{{\rm{d}}}{{{\rm{d}}a}}\left( {\dfrac{2}{3}\sqrt {{a^3}} } \right)\)(d) (i) Show that \(\displaystyle\int_0^a {{{\rm{e}}^x}{\rm{d}}x} = {{\rm{e}}^a} - 1\) (ii) Find \(\dfrac{{\rm{d}}}{{{\rm{d}}a}}\left( {{{\rm{e}}^a} - 1} \right)\)(e) (i) Show that \(\displaystyle\int_0^a {\sin \left( x \right)\,{\rm{d}}x} = 1 - \cos \left( a \right)\) (ii) Find \(\dfrac{{\rm{d}}}{{{\rm{d}}a}}\left( {1 - \cos \left( a \right)} \right)\)(f) (i) Using integration by parts, show that \(\displaystyle\int_0^a {\ln \left( x \right)\,{\rm{d}}x} = a\ln \left( a \right) - a\) (ii) Find \(\dfrac{{\rm{d}}}{{{\rm{d}}a}}\left( {a\ln \left( a \right) - a} \right)\)