

Integration¶

If $\frac{dy}{dx}=f(x)$, then $y$ is the function whose derivative is $f(x)$ and is called the derivative of $f(x)$ or indefinite integral of $f(x)$, denoted by $\int f(x)dx$
Similarly if $y=\int f(x)dx$, then $\frac{dy}{dx}=f(x)$
Since the derivative of a constant is zero, all indefinite integrals differ by an arbitrary constant.
The process of finding an integral is called integration.

General formulas of Integration¶

\[ \int x^ndx=\frac{x^{n+1}}{n+1}+c,\;(n\rlap{/}{=}-1)n\in Q \]

\[ \int (ax+b)^ndx=\frac{(ax+b)^{n+1}}{a(n+1)}+c,\;(n\rlap{/}{=}-1) \]

\[ \int cf(x)dx=c\int f(x)dx,\; (c\text{ is a constant}) \]

\[ \int \frac{dx}{x}=\ln x +c \]

\[ \int e^x dx=e^x+c \]

\[ \int e^{mx}dx=\frac{e^{mx}}{m}+c \]

\[ \int a^xdx=\frac{a^x}{\ln a}+c, \; a>0 \]

Integration by parts: $$ \int (uv)dx=u\int v dx-\int \left( \frac{du}{dx} \int v dx \right)dx $$