Question

^π _ -π∫x cos x dx plz answer​

Answer 1

Appropriate Question :-

Evaluate the following :

$\rm \: \displaystyle\int_{ - \pi}^{\pi} \: x \: cosx \: dx$

$\large\underline{\sf{Solution-}}$

Given integral is

$\rm \: \displaystyle\int_{ - \pi}^{\pi} \: x \: cosx \: dx$

We know from properties of definite integrals,

$\boxed{ \rm{ \: \begin{gathered}\begin{gathered}\bf\:\displaystyle\int_{ - a}^{a}f(x) dx= \begin{cases} &\sf{0 \: \: if \: f( - x) = - f(x)} \\ \\ &\sf{2\displaystyle\int_{0}^{a} \sf \: f(x)dx \: \: if \: f( - x) = f(x)} \end{cases}\end{gathered}\end{gathered}\: }}$

So, Consider

$\rm \: f(x) = x \: cosx$

So,

$\rm \: f( - x) = - x \: cos( - x)$

$\rm \: f( - x) = - x \: cosx$

$\rm\implies \:f( - x) \: = \: - \: f(x)$

Hence,

$\rm\implies \:\boxed{ \rm{ \:\rm \: \displaystyle\int_{ - \pi}^{\pi} \: x \: cosx \: dx \: = \: 0 \: \: }} \:$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: properties}}}} \\ \\ \bigstar \: \bf{\displaystyle\int_{0}^{a}f(x) \: dx = \displaystyle\int_{0}^{a}f(y) \: dy}\\ \\ \bigstar \: \bf{\displaystyle\int_{a}^{b}f(x) \: dx \: = \: - \displaystyle\int_{b}^{a}f(x) \: dx}\\ \\ \bigstar \: \bf{\displaystyle\int_{0}^{a}f(x) \: dx \: = \: \displaystyle\int_{0}^{a}f(a - x) \: dx}\\ \\ \bigstar \: \bf{\displaystyle\int_{a}^{b}f(x) \: dx \: = \: \displaystyle\int_{a}^{b}f(a + b - x) \: dx}\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}$