# Integrals Formulas and Notes

Properties of Integrals

\displaystyle \int kf(x) dx = k \int f(x) dx
\displaystyle \int -f(x) dx = - \int f(x) dx
\displaystyle \int f(x) \pm g(x) dx = \int f(x) dx \pm \int g(x) dx

Power Rule

\displaystyle \int x^{n} dx = \frac{1}{n+1} x^{n+1}+C
Example
\displaystyle \int 3x^{5} dx
\displaystyle = \frac{3x^{5+1}}{5+1}+C
\displaystyle = \frac{3x^{6}}{6}+C
\displaystyle = \frac{x^{6}}{2}+C

Substitution Rule

\displaystyle \int f \left( g(x) \right)g'(x) dx = \int f(u)du    , \displaystyle u=g(x)
Example
\displaystyle \int \frac{x}{\sqrt{1-x^{2}} } dx

\displaystyle u = 1-x^{2}
\displaystyle \frac{du}{dx} = -2x \rightarrow du=-2x
\displaystyle \rightarrow xdx= - \frac{du}{2}

\displaystyle \int \frac{x}{\sqrt{1-x^{2}} } dx = \int - \frac{1}{2 \sqrt{u} } du
\displaystyle = - \frac{1}{2} \int \frac{1}{\sqrt{u}} du
\displaystyle = - \frac{1}{2} (2 \sqrt{u})+C
\displaystyle = - \sqrt{u}+C
Remember that \displaystyle u = 1-x^{2}
\displaystyle - \sqrt{u}+C = - \sqrt{1-x^{2}}+C

\displaystyle \int \frac{x}{\sqrt{1-x^{2}} } dx = - \sqrt{1-x^{2}}+C

Integration by Parts

\displaystyle \int udv = uv - \int vdu
Example
\displaystyle \int x \sin(3x) dx
\displaystyle u = x, so \displaystyle du = 1 \cdot dx
\displaystyle dv = \sin(3x) dx. Integrate dv with respect to x:
\displaystyle v = - \frac{\cos(3x)}{3}
\displaystyle \int x \sin(3x) = uv - \int vdu
\displaystyle = (x) \left( - \frac{\cos(3x)}{3} \right) - \int \left( - \frac{\cos(3x)}{3} \right) (1 \cdot dx)
\displaystyle = - \frac{ x \cos(3x)}{3} + \int \frac{\cos(3x) }{3} dx
\displaystyle = - \frac{ x \cos(3x)}{3} + \frac{\sin(3x)}{9} + C

Exponential and Logarithm Functions

\displaystyle \int a^{x} = \frac{a^{x} }{\ln(a)} + C

Example
Find the integral of \displaystyle f(x) = 5^{x}
\displaystyle \int 5^{x} = \frac{5^{x} }{\ln(5)} + C
Example
Find the integral of \displaystyle f(x) = e^{x}
\displaystyle \int e^{x} = \frac{e^{x} }{\ln(e)} + C
\displaystyle = \frac{e^{x} }{1} + C
\displaystyle = e^{x} + C

\displaystyle \int \ln(x) = x \ln(x) - x + C

Trigonometric Functions

\displaystyle \int \cos(x) dx = \sin(x)+C

\displaystyle \int \sin(x) dx = - \cos(x)+C

\displaystyle \int \tan(x) dx = \ln \left( \left| \sec(x) \right| \right)+C

\displaystyle \int \sec(x) dx = \ln{\left |\tan{\left (x \right )} + \sec{\left (x \right )} \right |}+C

\displaystyle \int \csc(x) dx = x \mathrm{cosh^{-1}}{\left (x \right )} - \sqrt{x^{2} - 1} +C

\displaystyle \int \cot(x) dx = \ln | \sin(x) |+C

\displaystyle \int \sec^{2}(x) dx = \tan(x)+C

\displaystyle \int \sec(x) \tan(x) dx = \sec(x)+C

\displaystyle \int \csc^{2}(x) = - \cot(x)+C

\displaystyle \int \csc(x) \cot(x) = - \csc(x)+C

Inverse Trigonometric Functions

\displaystyle \int \cos^{-1}(x) dx = x \cos^{-1}(x) - \sqrt{1- x^{2}} +C

\displaystyle \int \sin^{-1}(x) dx = x \sin^{-1}(x) + \sqrt{1- x^{2}} +C

\displaystyle \int \tan^{-1}(x) dx = x \tan^{-1}(x) - \frac{1}{2} \ln(1+ x^{2}) +C

\displaystyle \int \sec^{-1}(x) dx = x \sec^{-1}(x) - \frac{ \sqrt{ 1 - \frac{1}{x^{2} } } x \ln( \sqrt{ x^{2}-1} + x) }{ \sqrt{ x^{2}-1} } +C

\displaystyle \int \cot^{-1}(x) dx = \frac{1}{2} \ln (x^{2}+1 ) + x \cot^{-1}(x) +C

\displaystyle \int \csc^{-1}(x) dx = x \csc^{-1}(x) + \ln(x + \sqrt{x^{2}+1}) +C

Hyperbolic Functions

\displaystyle \int \cosh(x) dx = \sinh(x)+C

\displaystyle \int \sinh(x) dx = \cosh(x)+C

\displaystyle \int \tanh(x) dx = \ln \left( \left| \cosh(x) \right| \right)+C

\displaystyle \int \sinh^{-1}(x) dx = x \mathrm{sinh^{-1}}{\left (x \right )} - \sqrt{x^{2} + 1}+C

\displaystyle \int \cosh^{-1}(x) dx = x \mathrm{sinh^{-1}}{\left (x \right )} - \sqrt{x^{2} + 1}+C

\displaystyle \int \tanh^{-1}(x) dx = x \mathrm{tanh^{-1}}{\left (x \right )} + \ln{\left (x + 1 \right )} - \mathrm{tanh^{-1}}{\left (x \right )}+C

\displaystyle \int sech(x) dx = \tan^{-1}| \sinh(x)| +C

\displaystyle \int csch(x) dx = \ln{\left | \tanh( \frac{x}{2}) \right |}+C

\displaystyle \int coth(x) dx = \ln | \sinh(x) |+C

\displaystyle \int sech^{2}(x) dx = \tanh(x)+C

\displaystyle \int sech(x) \tanh(x) dx = sech(x)+C

\displaystyle \int csch^{2}(x) = - coth(x)+C

\displaystyle \int csch(x) coth(x) = -csch(x)+C

Other Common Integrals

\displaystyle \int \frac{1}{a^{2}+x^{2}} dx = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right)+C

\displaystyle \int \frac{1}{x^{2}-a^{2}} dx = \frac{1}{2a} \ln \left( \frac{x-a}{x+a} \right) +C

\displaystyle \int \frac{1}{a^{2}-x^{2}} dx = \frac{1}{2a} \ln \left( \frac{x+a}{x-a} \right) +C

\displaystyle \int \frac{1}{1+x^{2}} dx = \tan^{-1} \left( x \right)+C

\displaystyle \int \frac{1}{a^{2}-x^{2}} dx = \sin^{-1} \left( \frac{x}{a} \right)+C

\displaystyle \int \frac{1}{ \sqrt{1-x^{2} } } dx = \sin^{-1} \left( x \right)+C

\displaystyle \int \frac{1}{ x \sqrt{x^{2}-a^{2} } } dx = \frac{1}{a} \sec^{-1} \left( \frac{x}{a} \right)+C

\displaystyle \int \frac{1}{ x \sqrt{x^{2}-1 } } dx = \sec^{-1} \left( x \right)+C

\displaystyle \int \frac{1}{ \sqrt{ x^{2}+a^{2} } } dx = \ln( x + \sqrt{x^{2}+a^{2}} ) +C

\displaystyle \int \frac{1}{ \sqrt{ x^{2}-a^{2} } } dx = \ln( x + \sqrt{x^{2}-a^{2}} ) +C

\displaystyle \int \sqrt{a^{2}-x^{2} } dx = \frac{x}{2} \sqrt{a^{2}-x^{2} } + \frac{a^{2} }{2} \sin^{-1} \left( \frac{x}{a} \right) +C

\displaystyle \int \sqrt{a^{2}+x^{2} } dx = \frac{x}{2} \sqrt{a^{2}+x^{2} } + \frac{a^{2} }{2} \sinh^{-1} \left( \frac{x}{a} \right) +C

\displaystyle \int \sqrt{x^{2}-a^{2} } dx = \frac{x}{2} \sqrt{x^{2}-a^{2} } - \frac{a^{2} }{2} \cosh^{-1} \left( \frac{x}{a} \right) +C