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Reduction formula sin nx cos mx




reduction formula sin nx cos mx

Let varphi(n)int cosn x site cadeau de noel a acheter dx, integrate by parts begincasescosn-1 x u cos x dx dvendcases.
Now integrating by parts: x n d ( e a x ) x n e a x e a x d ( x n ) x n e a x n e a x x n 1 d x, displaystyle beginalignedint xn, textd(eax) xneax-int eax.
Note: You can also show dx-fracsinn-1xcosm1xmn dx by splitting of sinnx and then following similar procedure.Displaystyle int xneax, textdx.!I n d x sin n a x displaystyle I_nint frac textdxsin nax!Now integrating by parts: e a x d ( x n 1 ) x n 1 e a x x n 1 d ( e a x ) x n 1 e a x a x n 1 e a x d x, displaystyle beginalignedint.Shifting indices back by 1 (so n 1 n, n n 1 n I n 1 x n e a x a I n, displaystyle nI_n-1xneax-aI_n!C ( m concours meubles 1 ) I m, n 1 x m 1 ( a x 2 b x c ) n 1 a ( m 2 n 3 ) I m 2, n b ( m n 2 ) I m 1, n displaystyle -c(m-1)I_m,nfrac.I n 1 ( a x 2 b x c ) n d x displaystyle I_nint frac 1(ax2bxc)n,textdx!A 2 I n a x n cos a x n x n 1 sin a x n ( n 1 ) I n 2 displaystyle a2I_n-axncos axnxn-1sin ax-n(n-1)I_n-2!Cosnx dx cosn-1x cos x dx Let u cosn-1x and dv/dx cos x So, du/dx -(n-1)cosn-2x sin x; v sin x cosnx dx -cosn-1 x sin x sin x (n-1)cosn-2x sin x dx cosn-1x sin x sin2 x (n-1)cosn-2x dx cosn-1x sin.I n, m a 2 I m 2, n I m 2, n 1 displaystyle I_n,ma2I_m-2,n-I_m-2,n-1!



Integral Reduction formula I n d x ( x 2 a 2 ) n displaystyle I_nint frac textdx(x2-a2)n!
Operatorname I(n,m) int sinn x cosm x, mathrm dx int sinn x cosm-1 x cos x, mathrm dx u sinn x cosm-1 xiff, mathrm dunsinn-1 dx,mathrm dv cos x,mathrm dxiff vsin x beginalign operatorname I(n,m) sinn1 x cosm-1 x-noperatorname dxtag1 sinn1 x cosm-1 x-noperatorname dxtag2.
I n -1 or, i n -2.I m, n x m 1 a ( 2 n m 1 ) ( a x 2 b x c ) n 1 b ( n m ) a ( 2 n m 1 ) I m 1, n c ( m 1 ).You only need that sin2 xcos2.I n d x ( p x q ) n a x b displaystyle I_nint frac textdx(pxq)nsqrt axb!I n, m d x x m ( x 2 a 2 ) n displaystyle I_n,mint frac textdxxm(x2-a2)n!From 1 ex sin(x)dx excos x excos x.





( n 1 ) J n sin a x a cos n 1 a x ( n 2 ) J n 2 displaystyle (n-1)J_nfrac sin axacos n-1ax(n-2)J_n-2!
30 Start by setting: I n cos n x.
I n, m x m ( x 2 a 2 ) n d x displaystyle I_n,mint frac xm(x2-a2)n,textdx!

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