Question

∫
1/4x²+11.dy/dx. please solve it fast ​

Answer 1

Solution:-

Given

$\sf \to \: \int \bigg( \dfrac{1}{4} {x}^{2} + 11 \bigg)dx$

Now

$\to \sf \int \dfrac{1}{4} {x}^{2} dx + \int11dx$

$\sf \to \: \dfrac{1}{4} \int {x}^{2} dx + 11x + c$

$\to \sf \dfrac{1}{4} \bigg( \dfrac{ {x}^{2 + 1} }{2 + 1} \bigg) + 11x + c$

$\sf \to \: \dfrac{1}{4} \times \dfrac{ {x}^{3} }{3} + 11x + c$

$\sf \to \: \dfrac{ {x}^{2} }{12} + 11x + c$

Answer

$\sf \to \: \dfrac{ {x}^{2} }{12} + 11x + c$

Some properties

$\sf \to \: \int {x}^{n} dx = \dfrac{ {x}^{n + 1} }{n + 1}$

$\to \rm \: \int \dfrac{1}{x} dx = log_{e} |x| + c$

$\sf \to \int {e}^{x} dx = {e}^{x} + c$

$\sf \to \: \int {a}^{x} = \dfrac{ {a}^{x} }{ log_{e}a} + c$

Answer 2

Refer to the attachment.