Basic Integration
Example 1:
$$\int x^7\, dx $$
Solution:
$$\int x^7\, dx $$
$$\because \int x^n\, dx = \frac{x^{n+1}}{n+1}+C $$
$$\therefore \int x^7\, dx = \frac{x^{7+1}}{7+1}+C $$
$$=\frac{x^8}{8}+C$$
Example 2:
$$\int 7x^5\, dx $$
Solution:
$$\int 7x^5\, dx =7\int x^5\, dx $$
$$\because \int x^n\, dx = \frac{x^{n+1}}{n+1}+C $$
$$\therefore \int 7x^5\, dx \, dx = 7\left(\frac{x^{5+1}}{5+1}\right) +C $$
$$=7\left(\frac{x^6}{6}\right)+C$$
$$=\frac{7x^6}{6}+C$$
Example 3:
$$\int (x^3+5x^2)\, dx $$
Solution:
$$\int (x^3+5x^2)\, dx $$
$$=\int x^3\, dx +\int 5x^2\, dx $$
$$=\int x^3\, dx +5\int x^2\, dx $$
$$\because \int x^n \, dx= \frac{x^{n+1}}{n+1}+C$$
$$\therefore \int (x^3+5x^2)\, dx =\frac{x^{3+1}}{3+1}+5\left(\frac{x^{2+1}}{2+1}\right)+C$$
$$=\frac{x^4}{4}+5\left(\frac{x^3}{3}\right)+C$$
$$=\frac{x^4}{4}+\frac{5x^3}{3}+C$$