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.
Answer
a. This one just requires thinking backwards.
b.
Example 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.
Answer
a. Substitution
b.
Example 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.
Answer
a.All it takes is a little backwards thinking.
b.
Example 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.
Answer
a. Parts
b. u = x2 and v' = e2x
Example 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.
Answer
a. All it takes is a little backwards thinking.
b.
Example 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.
Answer
a. Partial Fractions
Example 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.
Answer
a. Substitution (or thinking backwards).
b. u = 3x + 4 and
Example 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.
Answer
a. Substitution
b.
Example 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.
Answer
a. Substitution
b. u = 7x – 12
Example 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.
Answer
a. All this needs is thinking backwards.
b.
Example 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.
Answer
a. Partial fractions or substitution
b. If substitution, u = 2x2 – 5x – 3
Example 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.
Answer
a. Substitution
b.
Example 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.
Answer
a. Parts
b. u = 2x or u = x (if you move the 2 outside the integral) and v' = cos(3x + π)
Example 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.
Answer
a. Substitution or partial fractions.
b. For substitution, take u = x2 + 4x.
Example 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.
Answer
a. Parts
b. u = (lnx)2 and v' = 1
Example 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.
Answer
a. A little backwards thinking is all that's needed.
b.
or
Example 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.
Answer
a. All it takes is a little backwards thinking.
b.
Example 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.
Answer
a. Parts
b. u = x2 and v' = x(1 + x2)-3
Example 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.
Answer
a. Parts
b. u = 3x + 1 and v' = sin x
Example 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.
Answer
a. Partial Fractions
Example 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.
Answer
a. Substitution
b. u = e4x
Example 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.
