Matte D - del 2

Question 1

Standardintegral

∫x^p dx=

Answer 1

∫x^p dx = x^(p+1)/(p+1) +C (p≠-1)

Question 2

Standardintegral

∫1/x dx =

Answer 2

∫1/x dx = ln |x| + C

Question 3

Standardintegral

∫e^x dx =

Answer 3

∫e^x dx = e^x + C

Question 4

Standardintegral

∫sinx dx =

Answer 4

∫sinx dx = -cosx + C

Question 5

Standardintegral

∫cosx dx =

Answer 5

∫cosx dx = sinx + C

Question 6

∫tanx dx=

Answer 6

∫tanx dx = -ln |cosx| + C

Question 7

∫cotx dx =

Answer 7

∫cotx dx= ln |sinx| + C

Question 8

Standardintegral

∫1/(x^2+1) dx=

Answer 8

∫1/(x^2+1) dx= arctanx + C

Question 9

Standardintegral

∫1/(√(1-x^2))dx =

Answer 9

∫1/(√(1-x^2))dx = arcsinx + C

Question 10

∫lnx dx=

Answer 10

∫lnx dx= xlnx-x+C

Question 11

∫1+(tanx)^2 dx

Answer 11

∫1+(tanx)^2 dx = tanx + C

Question 12

∫(tanx)^2 dx

Answer 12

∫(tanx)^2 dx = tanx-x+C

Question 13

∫1+(cotx)^2 dx=

Answer 13

∫1+(cotx)^2 dx= -cotx + C

Question 14

∫(cotx)^2 dx

Answer 14

∫(cotx)^2 dx = -cotx-x+C

Question 15

∫arcsinx dx=

Answer 15

∫arcsinx dx= xarcsinx + √(1-x^2) + C

Question 16

∫arctanx dx=

Answer 16

∫arctanx dx= xarctanx-1/2 * ln(x^2+1) + C

Question 17

∫a^x dx=

Answer 17

∫a^x dx= a^x/lna + C , a>0, a≠1

Question 18

∫^a logx dx =

Answer 18

∫^a logx dx = x* ^alogx - x* ^aloge + C

Question 19

∫1/(√(x^2+1) dx =

Answer 19

∫1/(√(x^2+1)) dx = ln(√(x^2+1)+x) +C

Question 20

∫√(x^2+1) dx =

Answer 20

∫√(x^2+1) dx = 1/2 * x√(x^2+1) + 1/2 * ln(√(x^2+1)+x) + C

Question 21

Räkneregel

∫(f’(x)/f(x)) dx =

Answer 21

∫(f’(x)/f(x)) dx = ln |f(x)| + C

Question 22

Räkneregel

∫f(ax+b)dx =

Answer 22

∫f(ax+b)dx =1/a * F(ax+b) + C (a≠0)

Question 23

Räkneregel

∫( f(x)+g(x))dx=

Answer 23

∫( f(x)+g(x))dx= ∫f(x)dx + ∫g(x)dx + C

Question 24

Räkneregel

∫kf(x)dx=

Answer 24

∫kf(x)dx= k∫f(x)dx +C

Question 25

Partiell integration

∫f(x)g(x)dx =

Answer 25

∫f(x)g(x)dx = Fg - ∫F(x)g’(x)dx

Question 26

Variabelsubstitution

∫f(g(t))g’(t)dt=

Answer 26

∫f(g(t))g’(t)dt= ∫f(x)dx , x=g(t) och g injektiv