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equivalent to: int xneax, textdxfrac 1aleft(xneax-nint xn-1eax, textdxright)! Integral Reduction formula Indxxn(ax2bxc)displaystyle I_nint frac textdxxn(ax2bxc! For each function to be graphed, the calculator creates a JavaScript function, which is then evaluated in small steps in order to draw the graph. In, mdxxm(a2x2)ndisplaystyle I_n,mint frac textdxxm(a2-x2)n! I_nfrac 2xnsqrt axba(2n1)-frac 2nba(2n1)I_n-1! To supplement the example, the above can be used to evaluate the integral for (say) n 5; I5cos5xdx. Calculating lower indices: n5,quad I_5tfrac 15cos 4xsin xtfrac avenirbio code promo 45I_3, n3,quad I_3tfrac 13cos 2xsin xtfrac 23I_1, back-substituting: I1 cosxdxsinxC1,displaystyle because I_1 int cos x,textdxsin xC_1, I3 13cos2xsinx23sinxC2,C2 23C1,displaystyle therefore I_3 tfrac 13cos 2xsin xtfrac 23sin xC_2,quad C_2 tfrac 23C_1, I5 I_5 frac 15cos 4xsin xfrac 45leftfrac 13cos 2xsin xfrac. In, mxm(x2a2)ndxdisplaystyle I_n,mint frac xm(x2-a2)n,textdx!

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Now integrating by parts: beginalignedint xn, textd(eax) xneax-int eax, textd(xn) xneax-nint eaxxn-1,textdx, endaligned! Im, ncosmaxsinnaxdxdisplaystyle I_m,nint frac cos maxsin nax, textdx! I_n-frac sqrt axb(n-1)bxn-1-frac a(2n-3)2b(n-1)I_n-1! Then we back-substitute the previous results until we have computed. The main idea is to express an integral involving an integer parameter (e.g.

Variable of integration, integration bounds and more can be changed in ". (2n-1 4ac-b2)I_nfrac 12frac 2(2axb ax2bxc)n-frac 128a(n-1)I_n-frac 12! Indx(pxq)naxbdisplaystyle I_nint frac textdx(pxq)nsqrt axb! In other words, the reduction formula expresses the integral. I_m,nbegincasesfrac 1a(n-1)sin m-1axcos n-1axfrac mn-2n-1I_m,n-2-frac 1a(m-1)sin m-1axcos n-1axfrac mn-2m-1I_m-2,nendcases!