1. ∫(7x+4)⁹dx=1/7∫u⁹du=u¹⁰/70+C=1/70*(7x+4)¹⁰+C
u=7x+4
du=7dx
2. ∫(eˣ+e⁻ˣ)²dx=∫(e⁻²ˣ+e²ˣ+2dx)=∫e²ˣdx+∫e⁻²ˣdx+2∫1dx=e²ˣ/2-(1/2)*e⁻²ˣ+2x+C=1/2*(4x-e⁻²ˣ+e²ˣ)+C
3. ∫dx/(tgx+ctgx)=∫(cosx*sinx)/(cos²x+sin²x)dx=∫sinx*cosxdx=∫udu=u²/2+C=sin²x/2+C=-(1/2)*cos²x+C
u=sinx
du=cosxdx
4. ∫3/e³ˣdx=∫3e⁻³ˣdx=3∫e⁻³ˣdx=-∫eᵃda=-eᵃ+C=-e⁻³ˣ+C
a=-3x
da=-3dx
5. ∫dx/(x²-4)=∫-dx/(4(1-x²/4))=-1/4∫dx/(1-x²/4)=-1/2∫du/(1-u²)=-1/2*(-(1/2)*ln|1-u|+(1/2)*ln|1+u|+C=1/4(ln|2-x|+ln|2+x|)+C
u=x/2
du=1/2dx