1. Integration by Substitution
Section 6.2
Integration by Substitution

2. Do-Now: Homework quiz
Suppose that a point moves along some unknown curve y = f(x) in the xy-plane in such a way that at each point (x, y) on the curve, the tangent line has a slope x Find an equation for the curve given that it passes through the point (2, 1).

3. U-Substitution 𝑑 𝑑𝑥 ( 𝑢 𝑛+1 𝑛+1 ) = ??? 𝑑 𝑑𝑥 ( 𝑢 𝑛+1 𝑛+1 ) = 𝑢 𝑛 𝑑𝑢 𝑑𝑥
𝑑 𝑑𝑥 ( 𝑢 𝑛+1 𝑛+1 ) = ???
𝑑 𝑑𝑥 ( 𝑢 𝑛+1 𝑛+1 ) = 𝑢 𝑛 𝑑𝑢 𝑑𝑥
If we reverse this use of the chain rule, we can write…
(𝑢 𝑛 𝑑𝑢 𝑑𝑥 ) 𝑑𝑥= 𝑢 𝑛+1 𝑛+1 + C

4. Using the substitution method
We can use this method of changing variables to turn an unfamiliar integral into one that we can work with.
The goal is to replace one portion of the integral with u, and the remainder with 𝑑𝑢 𝑑𝑥 . To do the u- substitution method successfully, only constants can remain unaccounted for. Any variable (often x) must be switched to u.

5. example 𝑠𝑖𝑛 2 𝑥 cos 𝑥 𝑑𝑥 Let u = sin x. Then du/dx = cos x.
Now, either …
1. Solve for dx, substitute, and divide out the cos x, or...
2. Recognize that du = cos x dx and make that substitution.
The result: 𝑠𝑖𝑛 2 𝑥 cos 𝑥 𝑑𝑥 = 𝑢 2 𝑑𝑢
= 𝑢 𝐶 = 𝑠𝑖𝑛 3 𝑥 3 +𝐶 ……check your answer by taking the derivative.

6. What if there is a constant left over?
𝑥 ( 𝑥 2 +3) 50 𝑑𝑥
Let u = 𝑥 Then du/dx = 2x.
Method 1: Solve for dx, then substitute: dx = du/(2x).
𝑥 𝑢 50 ∙ 𝑑𝑢 2𝑥 = 𝑢 50 𝑑𝑢= 𝑢 50 𝑑𝑢
= 1 2 ∙ 𝑢 𝐶 = ( 𝑥 ) 𝐶

7. Method 2: “beach boys” 𝑥 ( 𝑥 2 +3) 50 𝑑𝑥
𝑥 ( 𝑥 2 +3) 50 𝑑𝑥
Let u = 𝑥 Then du/dx = 2x.
We know that du = 2x dx. We have the x and the dx needed to make the du, but “wouldn’t it be nice” if we had a 2 also.
Multiply the integral by 2 and 1/2. Use the 2 to complete the du, and bring the ½ outside the integral. Then complete the integration as you did before.

8. Additional examples 1. cos 𝑥 𝑥 𝑑𝑥 2. 𝑑𝑥 ( 1 3 𝑥 − 8) 5
𝑑𝑥 ( 1 3 𝑥 − 8) 5
3. Challenge: 𝑥 2 𝑥 −1 𝑑𝑥

9. 1. 𝑑𝑥 𝑠𝑖𝑛 2 (2𝑥) 2. tan 𝑥 𝑑𝑥 Using trig identities
Evaluate the following integrals by using trig identities first.
𝑑𝑥 𝑠𝑖𝑛 2 (2𝑥)
tan 𝑥 𝑑𝑥

10. Homework QUIZ A) y = ln x + 4 B) y = 3 ln x + e
C) y = 3 ln x + 1 D) y = ex + 4

11. U-substitution in definite integrals
Two methods for calculating the definite integral:
1. Perform the u-substitution, integrate, substitute back in for u, and evaluate at the given limits of integration.
2. Perform the u-substitution, integrate, change the limits of integration from x to u, and evaluate the function of u at the new limits of integration.

12. Example
Calculate the integral using both methods…
0 2 𝑥 ( 𝑥 2 +1) 3 𝑑𝑥

13. 2. Evaluate 0 𝜋/8 𝑠𝑖𝑛 5 2𝑥 𝑐𝑜𝑠2𝑥 𝑑𝑥
Additional examples
1. Evaluate 0 𝜋/4 cos 𝜋 −𝑥 𝑑𝑥
2. Evaluate 0 𝜋/8 𝑠𝑖𝑛 5 2𝑥 𝑐𝑜𝑠2𝑥 𝑑𝑥

14. AP MC PRactice

15. AP MC Practice

16. More ap mc practice

17. Evaluate the following indefinite integrals. 1. sec 4𝑥 tan 4𝑥 𝑑𝑥
Do-Now: homework quiz
Evaluate the following indefinite integrals.
sec 4𝑥 tan 4𝑥 𝑑𝑥
2. 𝑥 7 𝑥 𝑑𝑥

18. Homework quiz
Evaluate the following integral:
𝑥 sin 𝑥 −6 𝑑𝑥

19. Separable differential equations
A differential equation y’ = f(x, y) is separable if f can be expressed as a product of a function of x and a function of y.
𝑑𝑦 𝑑𝑥 =𝑔 𝑥 ℎ(𝑦)
To solve this differential equation….
1. Separate the variables : 1 ℎ(𝑦) 𝑑𝑦=𝑔 𝑥 𝑑𝑥.
2. Integrate both sides. The result is an implicit function.
3. Apply initial condition (if applicable).
4. Solve for y to get an explicit function (if desired).

20. Examples Solve the differential equation: 𝑑𝑦 𝑑𝑥 =−4𝑥 𝑦 2
Solve the initial value problem for the solution you just found if y(0) = 1.
Now try solving some of the differential equations from the slope fields worksheet to see if the solutions match the pictures.