Take $as$ given the following integral computations:

${\displaystyle \underset{-}{\overset{}{}}}xcos(x)dx={\displaystyle \underset{-}{\overset{}{}}}{x}^{2}sin(x)dx=0$

${\displaystyle \underset{-}{\overset{}{}}}xsin(x)dx=2,{\displaystyle \underset{-}{\overset{}{}}}{x}^{2}cos(x)dx=-4$

Let $C([-,])be$ the space $of$ continuous functions $on$ the interval $-,with$ inner

product

$($:$f(x),g(x)$:$)={\displaystyle \underset{-}{\overset{}{}}}f(x)g(x)dx$

Let $g(x)=\frac{1}{\sqrt[\phantom{2}]{}}(cos(x)+sin(x))inC([-,]),$ which $is$ a unit vector with respect $to$ this

inner product.

Find $pro{j}_{g(x)}(x^2+3x).$

$(a)2cos(x)+2sin(x)$

$(b)-4cos(x)+6sin(x)$

$(c)\frac{2}{\sqrt[\phantom{2}]{}}(cos(x)+sin(x))$

$(d)2cos(x)+3sin(x)$

Answer: $(a)$