Question

PLEASE SOLVE MY QUESTION!!!​

Answer 1

${\orange{ \boxed{ \boxed{ \begin{array}{cc} \bf \: \to \: given \: : \\ \\ I \: = \displaystyle \int \: \rm \: \frac{1}{2} {x}^{2} \: dx \\ \\ \underbrace{{\pink{ { \boxed{ \begin{array}{cc} \sf \: we \: know \: that \: : \\ \\ \sf \hookrightarrow \: \displaystyle \int \: \rm \: {x}^{n} \: dx = \frac{ {x}^{n + 1} }{ n+ 1} + c\end{array}}}}}}_{ \underbrace{ \sf \: apply \: this \: rule}_ { \downarrow}} \\ \\ \rm = \frac{1}{2} \displaystyle \int \: \rm \: {x}^{2} \: dx \\ \\ \rm = \frac{1}{2} \times \frac{ {x}^{2 + 1} }{2 + 1} + c \\ \\ \rm = \frac{1}{2} \times \frac{ {x}^{ 3 } }{3} + c \\ \\ \rm = \frac{ {x}^{3} }{6} + c \end{array}}}}}$

Answer of the attachment :

${\orange{ \boxed{ \boxed{ \begin{array}{cc} \bf \: \to \: given \: : \\ \\ I \: = \displaystyle \int \: \rm \: {x}^{ - 3} \: dx\\ \\ \underbrace{{\pink{ { \boxed{ \begin{array}{cc} \sf \: we \: know \: that \: : \\ \\ \sf \hookrightarrow \: \displaystyle \int \: \rm \: {x}^{n} \: dx = \frac{ {x}^{n + 1} }{ n+ 1} + c\end{array}}}}}}_{ \underbrace{ \sf \: apply \: this \: rule}_ { \downarrow}} \\ \\ \rm = \frac{ {x}^{ - 3 + 1} }{ - 3 + 1} + c \\ \\ \rm = \frac{ {x}^{ - 2} }{ - 2} + c \\ \\ \rm = - \frac{1}{2 {x}^{2} } + c \end{array}}}}}$