U-Substitution

Introduction to u-Substitution
$\cdot$ What is $u$ -Substitution?

$\cdot$ Exercise: Find $\int (5x^4-6x) \cos (x^5-3x^2)dx$ .
- How to pick " $u$ "?
- How to verify the final answer?

Integrate: Polynomial Functions
$\int-7x(6x^2+1)^{10}dx$
Integrate: Radical Functions
1. $\int\frac{x^6}{{^3}\sqrt{(8-5x^7)^2}}dx$
2. $\int\sqrt{6-3x}$ $dx$
Integrate: Exponential Functions
$\int e^{2x}dx$
Integrate: Logarithmic Functions
1. $\int \frac{(\ln x)^3}{x}dx$
2. $\int \frac{dx}{x \ln x}$
Integrate: Trigonometric Functions
1. $\int \sin ^3 x \cos x\; dx$
2. $\int \sec ^2 x(\tan x-1)^{100}\;dx$
Not-So-Obvious U-Substitution
1. $\int \sqrt{x^3-8}x^5dx$
2. $\int {^3}\sqrt{1+x^2}x^5dx$
3. $\int \frac{1+x}{1+x^2}dx$
4. $\int \cot x$ $dx$
Evaluate Definite Integrals in Two Methods
Evaluate: $\int_{-1}^{2} \sqrt{6-3x} dx$ $\int_{- 1}^{2} 6 - 3 x d x$
1. Introduction to definite integrals.
2. Method 1: evaluate the definite integral in terms of " $x$ ".
3. Method 2: evaluate the definite integral in terms of " $u$ ".
4. Method 1 VS. Method 2.
Evaluate Definite Integrals
Evaluate: $\int_{0}^{\frac{\pi}{3}} \frac{\sin \theta}{\cos ^2 \theta}d \theta$
Definite Integral: Does Not Exist (DNE)
Evaluate: $\int_{1}^{5} \frac{dx}{(x-3)^2}$