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### Lesson 5-5c

U-Substitution or The Chain Rule of Integration

Objectives

- Recognize when to try ‘u’ substitution techniques
- Solve integrals of algebraic, exponential, logarithmic, and trigonometric functions using ‘u’ substitution technique
- Use symmetry to solve integrals about x = 0 (y-axis)

Vocabulary

- Change of Variable – substitution of one variable for another in an integral (sort of reverse of the chain rule)
- Even Functions – when f(-x) = f(x); even functions are symmetric to the y-axis
- Odd Functions – when f(-x) = -f(x); odd functions are symmetric to the origin

Integrals of Symmetric Functions

- If a function, f(x), is even [f(-x) = f(x)], then its integral from –a to a is
- If a function, f(x), is even [f(-x) = -f(x)], then its integral from –a to a isbecause of signed area (above axis and below) cancel each other out.

a

a

a

∫

∫

∫

-a

0

-a

f(x) dx = 2 f(x) dx

f(x) dx = 0

Example 1

= ∫(2x + 1)² 2/2 dx

∫(2x + 1)² dx

If we let u = 2x +1then it becomes u² and du = 2dx

we are missing a 2 from dxso we multiple by 1 (2/2)

= ½ ∫(2x + 1)² 2 dx

½ goes outside ∫ and 2 stays with dx

= ½ ∫u² du = 1/6 u³ + C

= 1/6 (2x + 1)³ + C

Example Problems

∫

1) (2x +3) cos(x² + 3x) dx

Find the derivative of each of the following:

∫

- (5x² + 1)² (10x) dx

Let u = x² + 3x

then du = 2x + 3

So it becomes

cos u du

Let u = 5x² + 1

then du = 10x

So it becomes

u² du

∫

∫

= cos u du = sin (u) + C

= u² du = ⅓ u³ + C

= sin (x² + 3x) + C

= ⅓ (5x² + 1)³ + C

Example Problems cont

4

∫

3) tan²(t) cot (t) dt

Find the derivative of each of the following:

π/4

x

∫

t tan (t)dt

x

∫

6) √2 + sin (t) dt

2

Example Problems cont2x

∫

5) 5t sin (t) dt

Find the derivative of each of the following:

1

Example Problems cont

x³

∫

7) √1 + t4dt

Find the derivative of each of the following:

cos x

x

∫

8) t² dt

sin x

Summary & Homework

- Summary:
- alksdfj
- Homework:
- Day One: pg 420 - 422: 1, 2, 6, 8, 13, 21, 35, 42, 51, 58, 59, 76
- Day Two: pg 420 - 422: 1, 2, 6, 8, 13, 21, 35, 42, 51, 58, 59, 76

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