7 Integration

Question 1

integrate x^n

Answer 1

x^(n+1) /n+1 +c

Question 2

WHAT DON’T YOU FORGET

Answer 2

+C

Question 3

indefinite integral of e^x

Answer 3

e^x +c

Question 4

indefinite integral of cosx

Answer 4

sinx +c

Question 5

find the area of 5+ 2sin3x betweenx=0 and x=1/6pi

Answer 5

5/6pi +2/3

Question 6

when is there no indefinite integral for 1/x

Answer 6

when x is a negative

Question 7

why is y=ln(-x) possible

Answer 7

-x is positive when x values are negative

Question 8

y= ln(-x), dy/dx=

Answer 8

-(1/-x) = 1/x

Question 9

if x>0 and x<0 then

Answer 9

x (=/) 0

Question 10

5–>2 integrate for 3-2/x

Answer 10

3x-2ln|x|

9-2ln|x|

Question 11

3—>1 integrate (x^2 +1)/x

Answer 11

x^2/x +1/x

1/2x^2 +ln|x|

4+ln3

Question 12

integrate sqrrt(1-x^2)

Answer 12

u= 1-x^2

du/dx = -2x
du=-sxdx

S x(u^1/2)(-2x)dx
-1/2S(1-x^2)^1/2 dx

• 1/2 x2/3(1-x^2)^3/2 +c
• 1/3(1-x^2)^3/2 +c

Question 13

when you notice that the numerator is the derivative of the denominator

f’(x)/f(x)

Answer 13

du/dx= f’(x)

du= f’(x) dx

=ln|f(x)| +c

Question 14

S x^3/(X^4+1)dx

Answer 14

make so that it is f’(x)/f(x) —–> 1/4 S 4x^3/x^4+1 dx

1/4ln|x^4 +1| +c

Question 15

integration by parts

Answer 15

integrating backwards

uv+c = udv/dx + vdu/dx

uv+c =Sudv/dx + Svdu/dx

udv/dx = uv- S vdu/dx dx +c

Question 16

S xcos3x dx

Answer 16

u=x dv/dx =cos3x
du/dx= 1 v=1/3sin3x

x/3sin3x + S(1/3sin3x)(1)dx +c

x/3sin3x +1/9cos3x +c

Question 17

2—>0 S xe^3x dx

Answer 17

u= x dv/dx= e^3x
du/dx=1 v=1/3e^3x

x/3 e^3x - S1/3e^3x dx

2/3e^6 -(1/9e^6 -1/9e^0)

Question 18

two standard integrals

S 1/1+x^2

x=tanu

Answer 18

dx/du= sec^2u
dx=sec^2udu

1/sec^2u xsec^2udu =sec^2u/sec^2u du

S1du =u +c
x= tanu u= tan^-1x

S 1/(1+x^2) dx = tan^-1x +c

Question 19

u= 5x to find S 1/sqqrt(1-25x^2) dx

Answer 19

S sqqrt(1-u)/sqqrt(1-u) du

S1du= u+c = 5x +c

Question 20

S 1/ (a^2 +x^2) dx =

Answer 20

1/a tan^-1 (x/a) +c

S1/(sqrrt(a^2 -x^2) dx = sin^-1 (x/a) +c

Question 21

S 1/(9-x^2) dx

Answer 21

x = 3tanu

dx/du =3sec^2u

dx = 3sec^2u du

S 3sec^2u/9sec2u du
=S1/3du =1/3u +c

u= tan^-1(x/3)

1/3tan^-1(x/3) +c

Question 22

S 1/(Ax+b)

Answer 22

1/a ln|ax+b| +c

Question 23

S x/4+x^2 dx

Answer 23

1/2 S2x/(4+x^2) dx

1/2ln|4+x^2| +c

Question 24

S x/sqrrt(4-x^2) dx

Answer 24

u=4-x^2

du/dx =-2x

dx= -1/2x du

Sxx/sqrrt(4-x^2) x 1/2x du

S 1/2sqrrt(4-x^2) du

• 1/2 S u^-1/2 du
• 1/2 x 2u^1/2 = -sqrrt (4-x^2) +c