Aim: How do we integrate using u-­‐substitution? Get Ready: 2
1. f '(x) = 3x −1 ; f (2) = 4 f '(x) = 8x 3 + 5
2. f (1) = −4
1
f ''(x) =
x3
f '(4) = 2
3. f (0) = 0
f ''(x) = sin x
f '(0) = 1
4. f (0) = 6
I. U-­‐Substitution-­‐ Recognizing Patterns ∫ f '(g(x))g '(x)dx = f (g(x)) + C
u = g(x), du = g '(x)dx
∫ f '(u)du = f (u) + C
1. f '(x) = (x + 5) (3x ) à ∫ (x 3 + 5)100 (3x 2 )dx 3
100
2
2. f '(x) = (x 2 + 1)2 (2x) à ∫ (x 2 + 1)2 (2x)dx 3. f '(x) = 5 cos(5x) à ∫ 5 cos(5x)dx 3x
4. ∫
2
x 3 +1dx
Aim: How do we integrate using u-­‐substitution? −4 csc 2 (4x)dx
∫
5. 6. 2
3
∫ 6(6x −1) dx sin
7. ∫
3
x cos xdx
2
x(tan x + 3)dx
4x
1. ∫
3. 3
x 4 + 3 dx
2. sec
8. ∫
II. Additional Examples 4. ∫ sec
∫ (x
∫
2
2
x tan x dx
2x + 2
dx
+ 2x −1)2 −sin x
dx
cos x Aim: How do we integrate using u-­‐substitution? 3x 2 cos(x 3 )dx
∫
5. ∫ cos x(sin x + 3)
6. 3
4
dx
7. ∫ 9x 2 3x 3 + 2 dx
8. 9. ∫
∫
10. 10x 4 +1
3
2x 5 + x
dx
( x + 2)3
dx
2 x
∫ −2 csc
2
x(2 cot x + 3)3 dx