Basic Integration
Example 1:
$$\int x^7\, dx $$
Solution:
\begin{align*}
\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
\end{align*}
Example 2:
$$\int 7x^5\, dx $$
Solution:
\begin{align*}
\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\\
\end{align*}
Example 3:
$$\int (x^3+5x^2)\, dx $$
Solution:
\begin{align*}
\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\\
\end{align*}