Trigonometric Integrals–Solutions
Friday, January 23
Review
Compute the following integrals using integration by parts. It might be helpful to make a substitution.
1. Re2
1√xln(x)dx
4
9(1 + 2e3)
2. R1
0x√1 + x dx
4
15 (1 + √2)
Discuss: does the best strategy for solving each of the following integrals use substitution, integration by
parts, both, or neither?
1. Rxln(x)dx: IBP (u= ln x)
2. Rln(x)
xdx: sub u= ln x
3. R1
xln(x)dx: sub u= ln x
4. R1/x dx: neither
5. Rln(x)dx: IBP (u= ln x)
6. Rcos(x)esin(x)dx: sub u= sin x
7. Rx√1 + x dx: IBP u=x
8. Rx√x dx: niether (rewrite as x3/2)
9. Rsin(x) cos(x)esin(x)dx: sub u= sin x
Trig Formulas to Memorize:
1. sin2(x) + cos2(x) = 1
2. sin(2x) = 2 sin(x) cos(x)
3. cos(2x) = cos2(x)−sin2(x)
4. tan2(x) + 1 = sec2(x).
5. Rsin(x)dx =−cos(x) + C
6. Rcos(x)dx = sin(x) + C
7. Rsec2(x)dx = tan(x) + C
8. Rsec(x) tan(x)dx = sec(x) + C
Also Good to Know:
1. sin(a±b) = sin(a) cos(b)±cos(a) sin(b)
2. cos(a±b) = cos(a) cos(b)∓sin(a) sin(b)
3. cos(2x) = 2 cos2(x)−1
4. cos(2x)=1−2 sin2(x)
5. sin2(x) = (1 −cos(2x))/2
6. cos2(x) = (1 + cos(2x))/2
Formulas to Write on a Cheat Sheet:
Everything else.
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