Human Calculator

define $I_{a,b}= \int \cos^a{x}\sin{bx}dx$

$\begin{aligned}I_{a,b}&= \int \cos^a{x}\sin{bx}dx \\&= \int cos^{a-1}x(\cos{x}\sin{bx})dx \\&= \int \cos^{a-1}x(\sin{(b+1)x}+\sin{(b-1)x})dx\\&=\int \cos^{a-1}x\sin{(b+1)x}dx+\int \cos^{a-1}x\sin{(b-1)x}dx \\&= I_{a-1,b+1}+I_{a-1,b-1}\end{aligned}$

$I_{0,b}= \int \sin{bx}dx = -\frac{1}{b}\cos{bx}$

the problem is $I_{10,20}$

$\begin{aligned}I_{10,20}&=I_{9,19}+I_{9,21}\\&=I_{8,18}+2I_{8,20}+I_{8,22}\\&=I_{7,17}+3I_{7,19}+3I_{7,21}+I_{7,23}\\&= I_{6,16}+4I_{6,18}+6I_{6,20}+4I_{6,12}+I_{6,24}\\&=I_{5,15}+5I_{5,17}+10I_{5,19}+10I_{5,21}+5I_{5,23}+I_{5,25}\\&= I_{4,14}+6I_{4,16}+15I_{4,18}+20I_{4,20}+15I_{4,22}+6I_{4,24}+I_{4,26}\\&=I_{3,13}+7I_{3,15}+21I_{3,17}+35I_{3,19}+35I_{3,21}+21I_{3,23}+7I_{3,25}+I_{3,27}\\&=I_{2,12}+8I_{2,14}+28I_{2,16}+56I_{2,18}+70I_{2,20}+56I_{2,22}+28I_{2,24}+8I_{2,26}+I_{2,28}\\&=I_{1,11}+9I_{1,13}+36I_{1,15}+84I_{1,17}+126I_{1,19}+126I_{1,21}+84I_{1,23}+36I_{1,25}+9I_{1,27}+I_{1,29}\\&=I_{0,10}+10I_{0,12}+45I_{0,14}+120I_{0,16}+210I_{0,18}+252I_{0,20}+210I_{0,22}+120I_{0,24}+45I_{0,26}+10I_{0,28}+I_{0,30}\end{aligned}$