If there is one prayer that you should pray/sing every day and every hour, it is the LORD's prayer (Our FATHER in Heaven prayer)
It is the most powerful prayer. A pure heart, a clean mind, and a clear conscience is necessary for it.
- Samuel Dominic Chukwuemeka

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

## Standard Integrals of Exponents

$a, b, n \:\:are\:\:positive\:\:constants \\[3ex] (1.)\:\: \displaystyle\int ax^n dx = \dfrac{ax^{n + 1}}{n + 1} + C \\[7ex] (2.)\:\: \displaystyle\int (ax \pm b)^n dx = \dfrac{(ax \pm b)^{n + 1}}{a(n + 1)} + C \\[7ex] (3.)\:\: \displaystyle\int (-ax \pm b)^n dx = -\dfrac{(ax \pm b)^{n + 1}}{a(n + 1)} + C$

## Standard Integrals of Exponential Functions

$a, b, k, n \:\:are\:\:positive\:\:constants \\[3ex] (1.)\:\: \displaystyle\int a^x dx = \dfrac{a^x}{\ln a} + C \\[7ex] (2.)\:\: \displaystyle\int e^x dx = e^x + C \\[7ex] (3.)\:\: \displaystyle\int e^{kx} dx = \dfrac{e^{kx}}{k} + C \\[7ex] (4.)\:\: \displaystyle\int e^{-kx} dx = \dfrac{-e^{-kx}}{k} + C \\[7ex] (5.)\:\: \displaystyle\int e^{ax \pm b} dx = \dfrac{e^{ax \pm b}}{n} + C \\[7ex] (6.)\:\: \displaystyle\int k^{ax \pm b} dx = \dfrac{k^{ax \pm b}}{a\ln k} + C \\[7ex] (7.)\:\: \displaystyle\int xe^{ax} dx = \dfrac{e^{ax}(ax - 1)}{a^2} + C \\[7ex] (8.)\:\: \displaystyle\int x^ne^{ax} dx = \dfrac{x^ne^{ax}}{a} - \dfrac{n}{a}\displaystyle\int x^{n - 1}e^{ax} dx \\[7ex] (9.)\:\: \displaystyle\int \dfrac{dx}{1 + ke^{ax}} = x - \dfrac{\ln(1 + ke^{ax})}{a} + C$

## Standard Integrals of Logarithmic Functions

$n \:\:is\:\:a\:\:positive\:\:constant \\[3ex] (1.)\:\: \displaystyle\int \ln x dx = x\ln x - x + C \\[7ex] (2.)\:\: \displaystyle\int (\ln x)^n dx = x(\ln x)^n - n\displaystyle\int (\ln x)^{n - 1}dx \\[7ex] (3.)\:\: \displaystyle\int \dfrac{dx}{x\ln x} = \ln|\ln x| + C \\[7ex] (4.)\:\: \displaystyle\int x^n\ln x dx = \dfrac{x^{n + 1}[\ln x(n + 1) - 1]}{(n + 1)^2} + C$

## Standard Integrals of Trigonometric Functions

$(1.)\:\: \displaystyle\int \sin x dx = -cos x + C \\[7ex] (2.)\:\: \displaystyle\int cos x dx = sin x + C \\[7ex] (3.)\:\: \displaystyle\int \sec^2 x = \tan x + C$

## Standard Integrals of Hyperbolic Functions

$(1.)\:\: \displaystyle\int \sinh x dx = \cosh x + C \\[7ex] (2.)\:\: \displaystyle\int \cosh x dx = \sinh x + C$

## Standard Integrals of Rational Functions

$a, b, n \:\:are\:\:positive\:\:constants \\[3ex] (1.)\:\: \displaystyle\int \dfrac{1}{x} dx = \ln x + C \\[7ex] (2.)\:\: \displaystyle\int \dfrac{1}{ax \pm b} dx = \dfrac{\ln|ax \pm b|}{a} + C \\[7ex] (3.)\:\: \displaystyle\int \dfrac{1}{\sqrt{1 - x^2}} dx = \sin^{-1}x + C \\[7ex] (4.)\:\: \displaystyle\int \dfrac{-1}{\sqrt{1 - x^2}} dx = \cos^{-1}x + C \\[7ex] (5.)\:\: \displaystyle\int \dfrac{1}{1 + x^2} dx = \tan^{-1}x + C \\[7ex] (6.)\:\: \displaystyle\int \dfrac{1}{\sqrt{x^2 + 1}} dx = \sinh^{-1}x + C \\[7ex] (7.)\:\: \displaystyle\int \dfrac{1}{\sqrt{x^2 - 1}} dx = \cosh^{-1}x + C \\[7ex] (8.)\:\: \displaystyle\int \dfrac{1}{1 - x^2} dx = \tanh^{-1}x + C$

## Standard Integrals of Absolute Value Functions

$a, b, n \:\:are\:\:positive\:\:constants \\[3ex] (1.)\:\: \displaystyle\int |x| dx = \dfrac{x|x|}{2} + C \\[7ex] (2.)\:\: \displaystyle\int |ax \pm b| dx = \dfrac{(ax \pm b)|ax \pm b|}{2a} + C$

## Other Standard Integrals

$\underline{Algebraic\;\;Substitution} \\[3ex] (1.)\:\: \displaystyle\int f(x)f'(x) dx = \dfrac{f^2(x)}{2} + C \\[7ex] (2.)\:\: \displaystyle\int \dfrac{f'(x)}{f(x)} dx = \ln f(x) + C \\[7ex] (3.)\:\: \displaystyle\int \dfrac{-f'(x)}{f(x)} dx = -\ln f(x) + C \\[7ex] \underline{Trigonometric\;\;Substitution} \\[3ex] (4.)\;\; \displaystyle\int \dfrac{dx}{a^2 + x^2} = \dfrac{1}{a}\tan^{-1}\left(\dfrac{x}{a}\right) + C \\[7ex] (5.)\;\; \displaystyle\int \dfrac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\left(\dfrac{x}{a}\right) + C \\[7ex] (6.)\;\; \displaystyle\int \sqrt{a^2 - x^2}dx = \dfrac{a^2}{2}\left[\sin^{-1}\left(\dfrac{x}{a}\right) + \dfrac{x\sqrt{a^2 - x^2}}{a^2}\right] + C \\[7ex] \underline{Hyperbolic\;\;Substitution} \\[3ex] (7.)\;\; \displaystyle\int \dfrac{dx}{\sqrt{a^2 + x^2}} = \sinh^{-1}\left(\dfrac{x}{a}\right) + C \\[7ex] (8.)\;\; \displaystyle\int \dfrac{dx}{\sqrt{x^2 - a^2}} = \cosh^{-1}\left(\dfrac{x}{a}\right) + C \\[7ex]$