Calculo integrales ejercicios

Tarea 3.1
Tama: integración
Subtema: integración de formula directa
1.∫▒〖〖3x〗^4 dx=〗 6.∫▒〖(〖4x〗^3+x^2 )dx=4∫▒〖x^3 dx+∫▒〖x^2 dx=〗〗〗
3∫▒x^4 dx=〖3x〗^5/5+c 〖4x〗^4/4+x^3/3+c= x^4+1/3 x^3+c=
3/5 x^5+c

2. ∫▒1/x^3 dx=7.∫▒〖y^3 (〖2y〗^ -3)dy=∫▒〖〖2y〗^6-〖3y〗^3 dy〗〗
∫▒〖x^(-3) dx=x^(-2)/(-2)〗+c 〖2y〗^7/7-〖3y〗^4/4+c=2/7 y^7-3/4 y^4+c
-1/〖2x〗^2 +c

3. ∫▒〖5u^(3/2) 〗 du= 8.∫▒〖(3-2t+t^2 )dt=∫▒〖3dt-2∫▒〖tdt+∫▒t^2 〗〗〗
5∫▒u^(3/2) du=(5u^(5/2))/(5/2)+c 3t-〖2t〗^2/2+t^3/3+c=3t-1/2 t^2+1/3t^3+c
10/5 u^(5/2)+c=〖2u〗^(5/2)+c
4.∫▒2/√(3&x) dx= 9.∫▒(〖8x〗^4+〖4x〗^3-〖6x〗^2-4x+5)dx=
∫▒〖5x^(-1/3) 〗 dx=2∫▒x^(-1/3) dx 8∫▒〖x^4 dx+4∫▒〖x^3 dx-6∫▒〖x^2 dx-1∫▒〖xdx+∫▒5dx〗〗〗〗
〖2x〗^(2/3)/(2/3)+c=〖3x〗^(2/3)+c〖8x〗^5/5+〖4x〗^4/4-〖6x〗^3/3-〖4x〗^2/2+5x+c
( 〖8x〗^5)/5+〖4x〗^4/4-〖6x〗^3/3-〖4x〗^2/2+5x+c
8/( 5) x^5+x^4-〖2x〗^3-〖2x〗^2+5x+c
5.∫▒〖6t〗^2 √(3&t) dt=∫▒〖〖6t〗^2 t^(1/3) dt〗 10.∫▒〖√x (x+1)dx=∫▒x^(1/2) 〗 (x+1)dx
6∫▒〖t^(7/3) dt= 〖6t〗^(10/3)/(10/3)〗+c=18/10 t^(10/3)+c∫▒x^(3/2) +x^(1/2) dx=x^(5/2)/(5/2)+x^(3/2)/(3/2)+c
2/5 x^(5/2)+2/3 x^(3/2)+c
9/5 t^(10/3)+c
17.∫▒〖(4cscxcot+2〖sec〗^2 〗 x)dx
∫▒〖x^(3/2) dx-∫▒xdx=x^(5/2)/(5/2)-x^2/2+c〗 -4cscx+2tanx+c
2/5 x^(5/2)-1/2 x^2+c
18.∫▒〖(2〖cot〗^2 〗 θ-3〖tan〗^2 θ)dθ
∫▒〖〖2x〗^(-3)+〖3x〗^(-2)+5dx〗2(-x-cotx)-3(tanx-x)+c
2∫▒〖x^(-3) dx+3∫▒〖x^(-2) dx+∫▒〖5dx=〗〗〗 -3x-3cotx-tanx-3x+c
2^(-2)/2+3/(-1)+5x+c=1/〖-x〗^2 +3/(-x)+5x+c -6x-3cotx-3tanx+c

13.∫▒(x^2+4x-4)/√x dx= ∫▒(x^2-4x-4)/x^(1/2)
∫▒〖(x^2-4x-4)(x^(-1/2) )dx=∫▒〖x^(3/2)-〖4x〗^(1/2)-〖4x〗^(1/2) dx〗〗x^(5/2)/(5/2)-〖4x〗^(2/3)/(2/3)-〖4x〗^(1/2)/(1/2)+c=2/5 x^(5/2)-8/3 x^(3/2)-〖8x〗^(1/2)+c

14.∫▒〖(√(3&x)〗+1/√(3&x))dx= ∫▒x^(1/3) +x^(-1/3) dx
x^(4/3)/(4/3)+x^(2/3)/(2/3)+c= 4/4 x^(4/3)+3/2 x^(2/3)+c

15.∫▒(3sen-2cos)dt
-3 cos⁡〖- 2sen+c〗

16.∫▒(sen x)/(〖cos〗^2 x) dx=∫▒〖senx/cosx*1/cosx dx〗
∫▒〖secx tan〗 dx=secx+c

Tarea No. 3.2
Tama: integración
Subtema: técnica de integración por cambio de variables osustitución
1.∫▒〖√(1-4y) 〗 dy=∫▒〖(1-4y)〗^(1/2) (-4)/(-4) dy
u=1-4y -1/4 ∫▒u^(1/2) du
du=-4dy ( -1/4 u^(3/2))/(-3/2)+c= -1/6 (1-4y)^(3/2)+c
du/(-4)=dy

2.∫▒√(3&6-2x) dx= ∫▒〖〖(6-2x)〗^(1/3) (-2)/(-2) dx〗
u=6*2x -1/2 ∫▒u^(4/3) du
du=-2dx 〖-1/2 u〗^(4/3)/(4/3)+c=(-3)/8〖(6-2x)〗^(4/3)+c
3.∫▒〖x√(x^2-9 ) dx=∫▒〖〖(x〗^(2 )-9)^(1/2) 2/2 xdx〗〗
u=x^2-9 1/2 ∫▒u^(1/2) du
du=2x dx 〖1/2 u 〗^(3/2)/(3/2)+c= (-1)/3 〖〖(x〗^2-9)〗^(3/2)+c
4.∫▒〖x^2 (x^3-1)^10 〗 dx=∫▒〖(x^3-1)^10 〗 3/3 x^2 dx
u=x^3-1 1/3 ∫▒u^10 du
du= 〖3x〗^2 dx 〖1/3 u〗^11/11+c=1/33(x^3-1)^11+c4.∫▒〖x^2 (x^3-1)^10 〗dx = ∫▒〖(x^3-1)^10 3/3 x^2 dx 〗
u=x^3 -1 1/3 ∫▒u^10 du
du=〖3x〗^2 dx 〖1/3 u〗^11/11+c= 1/33(x^3-1)^(11 )+c

5.∫▒〖5x√(3&9-4x^2 )〗 )^2 dx=∫▒〖(9-〗 4x^2 )^(2/3) 5x dx
u=9-4x^2 5/(-8) ∫▒〖(9-4x^2 〗 )^(2/3)-8x/(-8) dx
-5/8(du=-8dx)...