$\displaystyle\displaystyle\frac{d}{dx}\displaystyle\int f(x)dx = f(x)$

\displaystyle\displaystyle\frac{d}{dx}\displaystyle\int f(x)dx = f(x)

$\displaystyle\int dx\,x^n=\displaystyle\displaystyle\frac{x^{n+1}}{n+1}$

\displaystyle\int dx\,x^n=\displaystyle\displaystyle\frac{x^{n+1}}{n+1}

$\displaystyle\int_a^b dx\,x^n=\displaystyle\displaystyle\frac{1}{n+1}(b^{n+1}-a^{n+1})$

\displaystyle\int_a^b dx\,x^n=\displaystyle\displaystyle\frac{1}{n+1}(b^{n+1}-a^{n+1})

$\displaystyle\int dx\,e^x=e^x$

\displaystyle\int dx\,e^x=e^x

$\displaystyle\int_a^b f(x)dx=\displaystyle\int_{u(a)}^{u(b)}f(u)\displaystyle\frac{dx}{du}du$

\displaystyle\int_a^b f(x)dx=\displaystyle\int_{u(a)}^{u(b)}f(u)\displaystyle\frac{dx}{du}du

$\displaystyle\int (f(x)+g(x))dx=\displaystyle\int f(x)dx+\displaystyle\int g(x)dx$

\displaystyle\int (f(x)+g(x))dx=\displaystyle\int f(x)dx+\displaystyle\int g(x)dx

$\displaystyle\int dx\displaystyle\displaystyle\frac{df}{dx}g(x)=f(x)g(x)-\int dx f(x)\displaystyle\displaystyle\frac{dg}{dx}$

\displaystyle\int dx\displaystyle\displaystyle\frac{df}{dx}g(x)=f(x)g(x)-\int dx f(x)\displaystyle\displaystyle\frac{dg}{dx}

$\displaystyle\int a\,dx=a\,x$

\displaystyle\int a\,dx=a\,x

$\displaystyle\int a\,f(x)dx=a\displaystyle\int f(x)dx$

\displaystyle\int a\,f(x)dx=a\displaystyle\int f(x)dx