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∫ dx/ (x+)= ln |x+ |+C

∫ dx/ (x2+2)= 1/a2

∫ dx/ (1+(x/)2)= 1/a arcsin (x/) +C

∫ sin kx dx= -1/k cos kx + C

∫(3x+5)dx 1/6(3x-5)2 + C

 

∫ xdx/ √(1+x2)=1/2 ∫ dx2/ √(1+u)=1/2 ∫ d(u+1)/ √1+u= ∫ dt/ √t= √t +C= √(1+u)+= √1+ x2 +

x2= u , 1+u=t

∫ sin3x cos x dx= ∫ t3 dt= t4 dt/4= (sin4 x/4)+C

Sin x=t, cos x dx= dt

∫(2x+1)/ (x2+x-3) dx = ∫d(x2+x-3)/ (x2+x-3)= ∫ du/u = ln |u|+= ln |x2+x-3| +C

U= (x2+x-3)

 

∫ dx/ 3 √(3x+1)2 = 1/3 ∫ du/ u2/3= u 1/3+C= 3 √(3x+1)+C

U=3x+1

du= 3 dx

dx= 1/3 du

 

∫ (1+x)/ (1+√x) dx

x=u2

Dx= 2udu

u=φ -1 (x) = √x

dx = 2∫ dx=2∫ (u2 u+2)du-4 ∫=2(1/3 u3-1/2x 2+2u)-4 ln (u+1)+C=2(1/3 x3/2-1/2x +2x1/2)- 4 ln(√x+1)+C

 


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