Question

integration from -π to π x-xsquare dx​

Answer 1

Given

$\to \sf \int_{ - \pi}^{ \pi}(x - {x}^{2})dx$

Now we can use this property

$\sf \to\int(ax \pm bx)dx = \int(ax)dx \pm \int(bx)dx$

So,

$\to\sf \int_{ - \pi}^{ \pi} (x)dx - \int_{ - \pi}^{ \pi} ( {x}^{2})dx$

We Know that

$\sf \to\int ({x}^{n})dx = \frac{ {x}^{n + 1} }{n + 1}$

By using this We get

$\sf \to \bigg[ \dfrac{x {}^{1 + 1} }{1 + 1} \bigg]_{ - \pi}^{ \pi} - \bigg[ \dfrac{x {}^{2 + 1} }{2 + 1} \bigg]_{ - \pi}^{ \pi}$

$\sf \to \bigg[ \dfrac{x {}^{2} }{2} \bigg]_{ - \pi}^{ \pi} - \bigg[ \dfrac{x {}^{3} }{3} \bigg]_{ - \pi}^{ \pi}$

$\sf \to \dfrac{1}{2} \bigg[ {x {}^{2} }{} \bigg]_{ - \pi}^{ \pi} - \dfrac{1}{3} \bigg[ {x {}^{3} }{} \bigg]_{ - \pi}^{\pi}$

$\sf \to\dfrac{1}{2} [(\pi) {}^{2} - ( - \pi) {}^{2} ] - \dfrac{1}{3} [(\pi) {}^{3} - ( - \pi) {}^{3} ]$

$\sf \to\dfrac{1}{2} [\pi^{2} - \pi^{2} ] - \dfrac{1}{3} [\pi^{3} + \pi{}^{3} ]$

$\sf \to\dfrac{1}{2} [0] - \dfrac{1}{3} [ 2\pi{}^{3} ]$

$\sf \to - \dfrac{1}{3} [ 2\pi{}^{3} ]$

Answer

$\sf \to - \dfrac{1}{3} [ 2\pi{}^{3} ]$