Lecture in Integration by Substitution

The Substitution Rule

Consider this example:

Now, you substitute u and the differential du into the integrand and integrate.

Lastly, re-substitute for u so the integral is in terms of the original variable, x.

How can you identify the correct substitution?

How do you know if you got the correct substitution?

Simple steps to determine the correct substitution

• Try to find good u in the integrand. It might be in parentheses or square root.
• Find the derivative of u, du/dx, and then by `cross-multiplying’ express dx in
  terms of du.
• Substitute all x variable with u. You may need to use some algebra to express the remaining x in terms of u, but commonly everything is in terms of u. 
•  Integrate in terms of u. After getting the integrals, replace u by the expression in x it stand for in the first step.

Integration by substitution with algebraic functions

For = , n -1

1.
Let u = x2 + 3x

Finding the differential we get, du = (2x + 3) dx

Substitute into the original problem, replacing all forms of x, getting

= =

Replacing back u by x2 + 3x

=

Integration by substitution involving exponential functions

       and       

1. I already give example for the first formula above. Here is for the second,

Let u = 3x

then, du = 3 dx,

⅓ du = dx

=  ⅓ =

Integration by substitution involving trigonometric functions

1. Integrate
Let u = 3x

then du = 3 dx,

⅓ du = dx

Substitute into the original problem, replacing all forms of x, getting

= 2 = 2sin(u) + C = 2sin3x + C

Algebraic substitution involving integral giving inverse trigonometric functions

  for a > 0

, for a > 0

, for u > a > 0

1.Integrate

Let u = tan x

then du = sec2x dx,

Substitute into the original problem, replacing all forms of x, getting

= =

Problems set exercises for Integration by Substitution

Follow this link to challenge yourself to work on different kinds of problems. Good luck. Integration by Substitution – Set 1 Problems

credit: D. A. Kouba (UC Davis), Renato E. Apa-ap, et al.©2013 www.PinoyBIX.com

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