Resoluciones Integrales

Integrales. Algunas resoluciones y respuestas
1) ∫ x 6 dx =
x7
+K
7
(K
cte)
2) ∫ (6 x 3 + 8 x 2 − 3) dx = ∫ 6 x 3 dx + ∫ 8 x 2 dx + ∫ −3dx = 6 ∫ x 3 dx + 8 ∫ x 2 dx − 3 ∫ dx =
=6
x4
x3
+8
− 3x + K
4
3
3
3
x 2
2
+K =
2x 2 + K
3) ∫ 2 x dx = ∫ (2 x) dx = 2 ∫ x dx = 2
3
3
2
4
7) 1 − e
8) 0
9)
3
4
10) A = 36
11) A =
3
1
12) A =
1
2
125
6
14) A =
15) ∫ ( x − 5) 4 dx = ∫ t 4 dt =
16) ∫ e
2 x +1
2
215
3
x−5 = t
dx = dt
t5
( x − 5) 5
+K =
+K
5
5
2x + 1 = t
1
1
1
1
dx = ∫ e dt = ∫ e t dt = e t + K = e 2 x +1 + K
2
2
2
2
2dx = dt
t
dx =
1
dt
2
1
17) ∫ 6 cos( 2 x − 1) dx =6 ∫ cos t dt2 = 6 ∫ cos tdt = 3sent + K = 3sen( 2 x − 1) + K
2
2x − 1 = t
1
2dx = dt dx = dt
2
7
18)
∫
x+3=t
dx = dt
7
t 2
2
( x + 3) dx = ∫ t 2 dt =
+ K = ( x + 3) 2 + K
7
7
2
5
5
u = 2x
21) ∫ 2 x e x dx = 2 xe x − ∫ 2e x dx = 2 xe x − 2e x + K
dv = e dx
x
u = ln x
1
22) ∫ ln x dx = x ln x − ∫ x dx = x ln x − x + K
x
23) ∫ x ln x dx = ln x
2
2
2
dv = x
2
x
x
x 1
1x
dx =
−∫
ln x −
+K
2
2 x
2
2 2
du = 2dx
v = ex
du =
v=x
1
dx
x
1
dx
x
x2
dv = xdx
v=
2
24) ∫ x senx dx = − x cos x − ∫ − cos xdx = − x cos x + ∫ cos xdx = − x cos x + senx + K
u = ln x
u=x
du =
du = dx
dv = senxdx
25)
∫ x cos x dx =
v = − cos x
xsenx − ∫ senxdx = xsenx − (− cos x) + K = xsenx + cos x + K
u=x
du = dx
dv = cos xdx
v = senx
x4
x4 1
x4
1
x4 x4
−∫
dx =
ln x − ∫ x 2 = ln x
−
+k
4
4 x
4
4
4 16
1
u = ln x
du = dx
x
x4
dv = x 3 dx
v=
4
26) ∫ x 3 ln x dx = ln x
sen3 x
sen3 x
sen3 x 1
−∫
dx = x
− ∫ sen3 xdx =
3
3
3
3
du = dx
27) ∫ x cos(3 x ) dx = x
u=x
sen3 x
3
sen3 x 1 ( − cos 3 x )
sen3 x 1 cos 3 x
=x
−
+K = x
+
+K
3
3
3
3
3 3
dv = cos 3 xdx
v=