Introduction to :

You've been learning the different methods of integration in a very artificial environment. You know that if you're in the "Integration by Substitution" section, you use substitution. If you're in the "Integration by Parts" section, you use integration by parts.

In the real world (by which we mean on exams), the directions probably won't say which method to use—you'll have to figure that out yourself.

The more you practice, the better you'll get at figuring out which method to use - you won't even have to think about it. In the meantime, we have some hints.

Practice:

Exercise 1

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 2

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 3

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 4

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 5

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 6

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 7

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 8

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 9

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 10

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 11

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 12

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 13

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 14

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 15

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 16

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 17

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 18

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 19

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 20

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 21

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


Exercise 22

For the integral, a. Determine whether the integral should be integrated by substitution, parts, partial fractions, or thinking backwards (no fancy techniques required).b. If you said 'substitution' for part (a), identify u. If you said `parts', identify u and v'. If you said `thinking backwards', go ahead and find the integral.


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