Documenti didattici di Matematica

Svolgimento: Applichiamo la semplice regola d'integrabilità  e otteniamo int_(1)^(2)(1/x-1/x^2-3x^2+12x)=[logx+1/x-x^3+6x^2]_(1)^(2)= log(2)+1/2-8+24-(log(1)+1-1+6)=log(2)+21/2 .

Svolgimento: int(x/(1+x))dx=int((1+x-1)/(1+x))dx= int(1)dx-int(1/(1+x))dx=x-log(|1+x|)+c .

Svolgimento: int((3x+2)/(4x+5))dx=1/4int((12x+8)/(4x+5))dx=1/4int((12x+15-7)/(4x+5))dx= =3/4int(1)dx-7/4int(1/(4x+5))dx= =3/4int(1)dx-7/(16)int(4/(4x+5))dx= 3/4x-7/(16)log(|4x+5|)+c .

Svolgimento: int_(1)^(3)(1/(x(1+x)))dx=int_(1)^(3)((1+x-x)/(x(1+x)))dx= =int_(1)^(3)(1/x)dx-int_(1)^(3)(1/(1+x))dx= [log(|x|)-log(|1+x|)]_(1)^(3)=log(1)-log(2)-(log(3)-log(4))=log(4)-log(3)+log(2) .

Svolgimento: int(1/(mx^2+n))dx=1/nint(1/(m/nx^2+1))dx= =1/nsqrt(n/m)int((sqrt(m/n))/((sqrt(m/n)x)^2+1))dx= =1/(sqrt(nm))arctg(sqrt(m/n)x)+c .

Svolgimento: La scomposizione in fattori è alquanto banale, infatti (x^2-3)=(x-sqrt3)(x+sqrt3) .

Svolgimento: (2a-b)^2+(b-2a)=(2a-b)^2-(2a-b)=(2a-b)(2a-b-1) .

Svolgimento: ax-ay+bx-by=a(x-y)+b(x-y)=(a+b)(x-y) .

Svolgimento: 3x-3y+x^2-xy=3(x-y)+x(x-y)=(3+x)(x-y) .

Svolgimento: 15x-6y-10x^2+4xy=3(5x-2y)-2x(5x-2y)=(3-2x)(5x-2y) .

Svolgimento: 15a^3+10a^2b-9a-6b=5a^2(3a+2b)-3(3a+2b)= =(5a^2-3)(3a+2b)=(sqrt5a+sqrt3)(sqrt5a-sqrt3)(3a+2b) .

Svolgimento: 3a^2-2ab-3a+2b=3a(a-1)-2b(a-1)=(3a-2b)(a-1) .

Svolgimento: 6a^2x-12abx-3a^2+6ab=3a^2(2x-1)-6ab(2x-1)=(3a^2-6ab)(2x-1)=3a(a-2b)(2x-1) .

Svolgimento: 10a^2b^2+20ab^3-2a^2b-4ab^2=10ab^2(a+2b)-2ab(a+2b)=(10ab^2-2ab)(a+2b)=2ab(5b-1)(a+2b) .

Svolgimento: 2xy+3y-2x-3=2x(y-1)+3(y-1)=(2x+3)(y-1) .

Svolgimento: 20ax-20bx+3ay-3by+5a-5b=a(20x+3y+5)-b(20x+3y+5)=(a-b)(20x+3y+5) .

Svolgimento: 6ax-3bx-4ay+2by-10a+5b=2a(3x-2y-5)-b(3x-2y-5)=(2a-b)(3x-2y-5) .

Svolgimento: 2a^2x+2a^2y-2a^2-6abx-6aby+6ab=2a^2(x+y-1)-6ab(x+y-1)= =(2a^2-6ab)(x+y-1)=2a(a-3b)(x+y-1) .

Svolgimento: Essendo a^2-b^2=(a+b)(a-b) , si ha 81x^4-1/(16)y^8=(9x^2+1/4y^4)(9x^2-1/4y^4) .

Svolgimento: 4x^3-xy^2=x(4x^2-y^2)=x(2x-y)(2x+y) .