5 Steps to a 5 AP Calculus BC 2019

436 Formulas and Theorems

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: sin^2 x=

1 −cos 2x

2

and

cos^2 x=

1 +cos(2x)

2

Note: After evaluating an integral, always

check the result by taking the derivative of

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

antiderivative).

14. The Fundamental Theorems 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).

15. 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)

]

16. Average Value of a Function:

f(c)=

1

b−a

∫b

a

f(x)dx

17. 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).

18. Area Bounded by 2 Curves:

Area=

∫x 2

x 1

(f(x)−g(x))dx,

where f(x)≥g(x).

19. Volume of a Solid with Known Cross Section:

V=

∫b

a

A(x)dx,

whereA(x)is the cross section.

20. Disc Method:

V=π

∫b

a

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

21. 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.

22. 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)

∣∣