U-Substitution

Lessons

• Introduction
  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?

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• 1.
  Integrate: Polynomial Functions
  $\int-7x(6x^2+1)^{10}dx$

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• 2.
  Integrate: Radical Functions
  a)

  $\int\frac{x^6}{{^3}\sqrt{(8-5x^7)^2}}dx$
  
  
  b)

  $\int\sqrt{6-3x}$ $dx$
  
  

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• 3.
  Integrate: Exponential Functions
  $\int e^{2x}dx$

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• 4.
  Integrate: Logarithmic Functions
  a)

  $\int \frac{(\ln x)^3}{x}dx$
  
  
  b)

  $\int \frac{dx}{x \ln x}$
  
  

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• 5.
  Integrate: Trigonometric Functions
  a)

  $\int \sin ^3 x \cos x\; dx$
  
  
  b)

  $\int \sec ^2 x(\tan x-1)^{100}\;dx$
  
  

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• 6.
  Not-So-Obvious U-Substitution
  a)

  $\int \sqrt{x^3-8}x^5dx$
  
  
  b)

  $\int {^3}\sqrt{1+x^2}x^5dx$
  
  
  c)

  $\int \frac{1+x}{1+x^2}dx$
  
  
  d)

  $\int \cot x$ $dx$
  
  

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• 7.
  Evaluate Definite Integrals in Two Methods
  Evaluate: $\int_{-1}^{2} \sqrt{6-3x} dx$
  a)

  Introduction to definite integrals.
  
  
  b)

  Method 1: evaluate the definite integral in terms of "$x$".
  
  
  c)

  Method 2: evaluate the definite integral in terms of "$u$".
  
  
  d)

  Method 1 VS. Method 2.
  
  

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• 8.
  Evaluate Definite Integrals
  Evaluate: $\int_{0}^{\frac{\pi}{3}} \frac{\sin \theta}{\cos ^2 \theta}d \theta$

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• 9.
  Definite Integral: Does Not Exist (DNE)
  Evaluate: $\int_{1}^{5} \frac{dx}{(x-3)^2}$

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