Integration formulas list | Proveiff Study

Introduction : In this list, we present integration formulas list of every type of mathematical functions and those are commonly used in all of the branches of study.

Formula of Integration of a real variable function f(x) is defined as follows

\[\int f(x) dx = F(x)+c\]

where c is an integration constant.

Integration of Algebraic functions

\(\int 0 dx =c\)
\(\int 1 dx =x+ c\)
\(\int k dx =kx+c\)
\(\int x dx =\frac{x^2}{2}+c\)
\(\int kx dx =k\int x dx= k\frac{x^2}{2}+c\) for any constant k.
\(\int (kx+l) dx =k\frac{x^2}{2}+lx+c\) for any constants k and l.
\(\int (x^{r}) dx =\frac{x^{r+1}{r+1}+c, r\ne -1\)
\(\int (\frac{1}{x}) dx =log(x)+c\)
\(\int (\sqrt{x}) dx =\frac{2x\sqrt{x}}{3}+c\)
\(\int (\frac{1}{\sqrt{x}}) dx=2 \sqrt{x}+c\)

Integration of Exponential functions

\(\int e^x dx=e^x+c\)
\(\int e^{lx+m} dx=\frac{e^{l x+m}}{l}+c\) where l, m are any constants.
\(\int e^{-x} dx=-e^{-x}+c\)
\(\int a^x dx=\frac{a^x}{\log (a)}+c\)
\(\int a^{-x} dx=-\frac{a^{-x}}{\log (a)}+c\)

Integration of Logarithmic functions

\(\int \log_e (x) dx=x \log_e (x)-x+c\)
\(\int \log_e (lx+m) dx=\frac{(l x+m) \log_e (l x+m)}{l}-x+c\)
\( int \log_a (x) =\frac{x \log_a (x)-x}{\log_e (a)}+c\)
\( int \log_a (lx+m) =\frac{\frac{(l x+m) \log_a (l x+m)}{l}-x}{\log_e (a)}+c\)

Integration of Trigonometric functions

\(\int \sin (x)dx=-\cos (x)+c\)
\( \int \cos (x) dx= \sin (x)+c\)
\( \int \tan (x) dx=-\log (\cos (x)) +c\)
\( \int \csc (x) dx=\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right) +c\)
\(\int \sec (x) dx =\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right) +c\)
\( \int \cot (x) dx=\log (\sin (x)) +c\)

Integration of Inverse Trigonometric functions

\(\int \sin ^{-1}(x)dx=\sqrt{1-x^2}+x \sin ^{-1}(x)+c\)
\(\int \cos ^{-1}(x) dx=x \cos ^{-1}(x)-\sqrt{1-x^2}+c\)
\(\int \tan ^{-1}(x) dx=x \tan ^{-1}(x)-\frac{1}{2} \log \left(x^2+1\right)+c\)
\(\int \csc ^{-1}(x) dx=\frac{\sqrt{x^2-1} \left(\log \left(\frac{x}{\sqrt{x^2-1}}+1\right)-\log \left(1-\frac{x}{\sqrt{x^2-1}}\right)\right)}{2 \sqrt{1-\frac{1}{x^2}} x}+x \csc ^{-1}(x)+c\)
\(\int \sec ^{-1}(x)dx=x \sec ^{-1}(x)-\frac{\sqrt{x^2-1} \left(\log \left(\frac{x}{\sqrt{x^2-1}}+1\right)-\log \left(1-\frac{x}{\sqrt{x^2-1}}\right)\right)}{2 \sqrt{1-\frac{1}{x^2}} x}+c\)
\(\int \cot ^{-1}(x) dx=\frac{1}{2} \log \left(x^2+1\right)+x \cot ^{-1}(x)+c\)

Integration of Trigonometric Hyperbolic functions

\(\int \sinh (x) dx=\cosh (x)+c\)
\(\int \cosh (x) dx=\sinh (x)+c\)
\(\int \tanh (x) dx=\log (\cosh (x))+c\)
\(\int \text{csch}(x) dx=\log \left(\tanh \left(\frac{x}{2}\right)\right)+c\)
\(\int \text{sech}(x) dx=2 \tan ^{-1}\left(\tanh \left(\frac{x}{2}\right)\right)+c\)
\(\int \coth (x) dx=\log (\sinh (x))+c\)

Integration of Trigonometric Inverse Hyperbolic functions

\(\int \sinh ^{-1}(x) dx=x \sinh ^{-1}(x)-\sqrt{x^2+1}+c\)
\(\int \cosh ^{-1}(x)dx=x \cosh ^{-1}(x)-\sqrt{x-1} \sqrt{x+1}+c\)
\(\int \tanh ^{-1}(x) dx=\frac{1}{2} \log \left(1-x^2\right)+x \tanh ^{-1}(x)+c\)
\(\int \text{csch}^{-1}(x) dx=x \left(\frac{\sqrt{\frac{1}{x^2}+1} \sinh ^{-1}(x)}{\sqrt{x^2+1}}+\text{csch}^{-1}(x)\right)+c\)
\(\int \text{sech}^{-1}(x) dx=x \text{sech}^{-1}(x)-\frac{\sqrt{\frac{1-x}{x+1}} \sqrt{1-x^2} \sin ^{-1}(x)}{x-1}+c\)
\(\int \coth ^{-1}(x) dx=\frac{1}{2} \log \left(1-x^2\right)+x \coth ^{-1}(x)+c\)