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

MA 3972-MA-Book May 24, 2018 14:33

406 Appendix

f.

∫

√^1

a^2 −x^2

dx=sin−^1

(x

a

)

+C

g.

∫

1

a^2 +x^2

dx=

1

a

tan−^1

(x

a

)

+C

h.

∫

1

x

√

x^2 −a^2

dx=

1

a

sec−^1

∣∣

∣

x

a

∣∣

∣+Cor

1

a

cos−^1

∣∣

∣

a

x

∣∣

∣+C

i.

∫

sin^2 xdx=

x

2

−

sin( 2 x)

4

+C

Note that sin^2 x=

1 −cos 2x

2

and

cos^2 x=

1 +cos^2 x

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).

14. Intergration by parts

∫

udv=uv−

∫

vdu

(and follow LIPET Rule).

15. The Fundamental Theorem of Calculus
    ∫b

a

f(x)dx=F(b)−F(a),

whereF′(x)=f(x).

IfF(x)=

∫x

a

f(t)dt, thenF′(x)=f(x).

16. Trapezoidal Approximation:
    ∫b

a

f(x)dx

=

b−a

2 n

[

f(x 0 )+ 2 f(x 1 )+ 2 f(x 2 )...

+ 2 f(xn− 1 )+f(xn)

]

17. Average Value of a Function:

f(c)=

1

b−a

∫b

a

f(x)dx

18. Mean Value Theorem:

f′(c)=

f(b)− f(a)

b−a

for somecin (a,b).

Mean Value Theorem for Integrals:

∫b

a

f(x)dx= f(c)(b−a) for somec

in (a,b).

19. Area Bounded by 2 Curves:

Area=

∫x 2

x 1

(f(x)−g(x))dx,

where f(x)≥g(x).

20. Volume of a Solid with Known Cross Section:

V=

∫b

a

A(x)dx,

whereA(x)is the cross section.

21. Disc Method:

V=π

∫b

a

(f(x))^2 dx, wheref(x)=radius.

22. Using the Washer Method:

V=π

∫b

a

(

(f(x))^2 −(g(x))^2

)

dx,

where f(x)=outer radius and

g(x)=inner radius.

23. Distance Traveled Formulas:
    
    
    • Position Function:s(t);s(t)=
    
    
    
    

∫

v(t)dt

• Velocity:v(t)=ds
  dt
  ;v(t)=

∫

a(t)dt

• Acceleration:a(t)=dv
  dt


• Speed:|v(t)|


• Displacement fromt 1 tot 2 =

∫t 2

t 1

v(t)

=s(t 2 )−s(t 1 ).