We've been learning the different methods of integration in a very artificial environment. We know that if we're in the "Integration by Substitution" section, we use substitution. If we're in the "Integration by Parts" section, we use integration by parts.
In the real world (by which we mean on exams), the directions probably won't say which method to use—we'll have to figure that ourselves.
The more we practice, the better we'll get at figuring out which method to use. We won't even have to think about it. In the meantime, we have some hints.
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.
