Calculus for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version

Activity 5.5.1.

1. ${\displaystyle u=16-9x^2}$
2. ${\displaystyle u=9x^2}$
3. ${\displaystyle u=3x}$
4. ${\displaystyle u=x}$

Activity 5.5.2.

1. b.
2. c.
3. g.
4. h.

Activity 5.5.3.

Activity 5.5.5.

1. Choosing $u^2=49x^2$ and $a^2=4\text{.}$
2. Finding $u=7x\text{,}$ $du=7dx\text{,}$ ${\displaystyle \frac{1}{7}du=dx\text{,}}$ and $a=2\text{.}$
3. Substituting ${\displaystyle \frac{3}{x\sqrt{49x^2-4}}dx}$ with ${\displaystyle 3\int \frac{1}{u\sqrt{u^2-a^2}}du}$ and finding the best match of f from Appendix A.
4. Integrating ${\displaystyle 3\int {\displaystyle \frac{1}{u\sqrt{u^2-a^2}}}du=3\left(\frac{1}{a}\mathrm{arcsec}\left({\displaystyle \frac{u}{a}}\right)\right)+C\text{.}}$
5. Unsubstituting ${\displaystyle 3\left(\frac{1}{a}\mathrm{arcsec}\left(\frac{u}{a}\right)\right)+C}$ to get ${\displaystyle \frac{3}{2}}\mathrm{arcsec}\left({\displaystyle \frac{7x}{2}}\right)+C\text{.}$

Activity 5.5.6.

Activity 5.5.7.

1. We prefer the substitution choice of $u^2=x^2$ and $a^2=64\text{.}$
2. We prefer the substitution choice of $u^2=9x^2$ and $a^2=64\text{.}$
3. We do not have a strong preference, since these substitution choices are of the same difficulty.

Activity 5.5.8.

Activity 5.5.9.

Activity 5.5.10.