Integration by Substitution
Example 1:
$$\int (x^2+1)^4 \, xdx $$Solution:
- $$\textup{Let}\,\, u=x^2+1$$
- $$du=2xdx$$
- $$\frac{du}2=xdx$$
- $$\int (x^2+1)^4 \, xdx = \int u^4\,\, \frac{du}2$$
- $$= \frac{1}2 \int u^4\,du$$
- $$= \frac{1}2 \, ( \frac{u^{4+1}}{4+1})+C$$
- $$= \frac{1}2 \, ( \frac{u^{5}}{5})+C$$
- $$= \frac{1}{10} \, u^5+C$$
- $$= \frac{1}{10} \, (x^2+1)^5+C$$