5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 8, 2018 13:52

Integration 229

Differentiation Formulas (cont.): Integration Formulas (cont.):

10.

d

dx

(lnx)=

1

x

10.

∫

1

x

dx=ln|x|+C

11.

d

dx

(ex) =ex 11.

∫

exdx=ex+C

12.

d

dx

(ax) =(lna)ax 12.

∫

axdx=

ax

lna

+Ca>0,a= 1

13.

d

dx

(sin−^1 x)=

√^1

1 −x^2

13.

∫

√^1

1 −x^2

dx =sin−^1 x+C

14.

d

dx

(tan−^1 x)=

1

1 +x^2

14.

∫

1

1 +x^2

dx =tan−^1 x+C

15.

d

dx

(sec−^1 x)=

1

|x|

√

x^2 − 1

15.

∫

1

|x|

√

x^2 − 1

dx=sec−^1 x+C

More Integration Formulas:

16.

∫

tanxdx=ln|secx|+Cor −ln|cosx|+C

17.

∫

cotxdx=ln|sinx|+Cor −ln|cscx|+C

18.

∫

secxdx=ln|secx+tanx|+C

19.

∫

cscxdx=ln|cscx−cotx|+C

20.

∫

lnxdx=xln|x|−x+C

21.

∫

1

√

a^2 −x^2

dx =sin−^1

(x

a

)

+C

22.

∫

1

a^2 +x^2

dx =

1

a

tan−^1

(x

a

)

+C

23.

∫

1

x

√

x^2 −a^2

dx=

1

a

sec−^1

∣

∣∣x

a

∣

∣∣+Cor^1

a

cos−^1

∣

∣∣a

x

∣

∣∣+C

24.

∫

sin^2 xdx=

x

2

−

sin(2x)

4

+C.Note: sin^2 x=

1 −cos 2x

2

Note that after evaluating an integral, always check the result by taking the derivative of the

answer (i.e., taking the derivative of the antiderivative).

TIP • Remember that the volume of a right-circular cone isv=^1
3
πr^2 hwhereris the radius
of the base andhis the height of the cone.