Integration by Substitution
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It is possible to transform a difficult integral to an easier integral in a different variable using a substitution.
By using substitutions, we can show that:

Example:
Find the integral of:
(a) sin x cos²x
(b)    3x²  
      x³ + 1

(a) Using the first of the two above formulae above, imagine f(x) = cos x. Therefore [f(x)]² = cos²x and f '(x) = sin x. Therefore, since n = 2, the answer is simply (cos³x)/ 3 + c

(b) Since the top is the differential of the bottom, we can use the second of the two formulae above to get the answer of  ln(x³ + 1) + c.

Using a Substitution
Sometimes you will be told to integrate a function by using a substitution.

© Matthew Pinkey

Other Notes in this Category

1. Chain, Product and Quotient
2. Differential Equations
3. Differentiation
4. Differentiation from 1st Principles
5. Differentiation of Trig Funcns
6. Exponentials & Logarithms
7. Implicit Differentiation
8. Integration
9. Integration by Parts
10. Integration by Substitution
11. Tangents and Normals
12. The Second Derivative
13. Uses of Differentiation
14. Volumes of Revolution

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