1. Michelle Owsley
Tahniyat Farooqi
Cyra Byramji

2. Derivative Anti-derivatives
sin(x) -cos(x)
cos(x) sin(x)
sec2(x) tan(x)
csc2(x) -cot(x)
sec(x)tan(x) sec(x)
csc(x)cot(x) -csc(x)
Xn+1
Xn
n+1
f’(g(x))g’(x) f(g(x))
f’(x) is also the same as the integral symbol ∫

3.  Derive: f(x)=3x2+2x
 Answer: f’(x)= 6x+2
 Now Anti-Derive: ∫6x+2dx
 Answer: f(x)=3x2+2x+c
 Where did the “c” come from?
 The “c” means any constant. When you derive a constant
in a function, the derivative is 0,so when anti-deriving
always add “+c” at the end because you cannot assume
whether or not there was a constant in the original
function, and by adding “c” you are making sure you
didn’t leave any numbers out of the function.
RULES

4.  Anti-derive the following:
 ∫sin(x)dx
  -cos(x) + c
 ∫csc2(X)dx
  -cot(x) + c
 ∫sec(x)tan(x)dx
  sec(x) + c
 ∫4x + cos(x)dx
  2x2 + sin(x) + c
 Is there a rule for “4x” (one something to a power)? YES! If
you don’t remember click the button
 4x= 4x2 = 2x2
2
RULES

5.  ∫3x2 - 7x + 4 - 5sec2(x)dx
 Step by Step
If you forgot your rules…
 Rule for 3x2? Yes  x3
 Rule for 7x? Yes  7x2
2
 Rule for 4? Yes  4x
 Rule for 5sec2(x)? Yes  5tan(x)
 Put it all together…
 X3 - 7x2 + 4x -5tan(x) + c
2
RULES

6.  If the function u=g(x) has a continuous derivative on the closed
interval [a,b] and f is continuous on the range of g, then
 ∫ab f(g(x))g’(x)dx = ∫g(a)g(b) f(u)du
 Anti-derive:
 ∫(x2+4) 9 (2x)dx
 Is there are rule for this one? Of course, it’s the Product
Rule…WRONG! If this is what you were thinking
then Click!
 You have to use u-substitution to solve this problem.
Derivative
 u= x2+4 and du=2xdx
 Now Substitute!

7.  u= x2+4
 du=2xdx
 ∫(x2+4) 9 (2x)dx
 ∫(u) 9du  This is your new equation, now is there a rule for
this? Yes, so Anti-derive.
 u 10 + c Not finished yet…now plug back in your original numbers.
10
 (x2+4)10 + c  This is the final answer
10
RULES

8.  ∫5cos(5x)dx
 u=5x  du=5dx
 ∫cos(u)du
 sin(u) + c
 sin(5x) + c
 ∫x(5x 2 - 3)7dx
 u=5x 2 – 3  du=10xdx but there is not 10x in the problem?!?
 That’s ok, 10 is a coefficient so just move it! 1/10du=xdx
 Now that we have everything…rewrite the problem
 ∫u7 (1/10)du
 (1/10)u8  u8 + c
8 80
 Don’t forget to plug the original numbers back in …
 (5x 2 - 3)8 + c
80
RULES