Integrals: Problems with Solutions
By Prof. Hernando Guzman Jaimes (University of Zulia - Maracaibo, Venezuela)Integral Formulas
$\int kdx=kx+C$ $k \in R$$\int x^n dx=\frac{1}{n+1}x^{n+1}+C$ $n \ne -1$ $n \in Z$
$\int \frac{1}{x} dx=ln(x)+C$
$\int e^x dx=e^x+C$
$\int a^x dx=\frac{a^x}{ln(a)}+C$ $a \in R, a > 0$
$\int \sin(x) dx=-\cos(x)+C$
$\int \cos(x) dx=\sin(x)+C$
$\int \sec^2(x) dx=\tan(x)+C$
$\int \csc^2(x) dx=-\cot(x)+C$
$\int \sec(x)\tan(x) dx=\sec(x)+C$
$\int \csc(x)\cot(x) dx=-\csc(x)+C$
$\int \tan(x) dx=\ln(\sec(x))+C$
$\int \cot(x) dx=\ln(\sin(x))+C$
$\int \sec(x) dx=\ln(\sec(x) + \tan(x))+C$
$\int \csc(x) dx=\ln(\csc(x) - \cot(x))+C$
$\int \frac{dx}{\sqrt{a^2-x^2}} dx=\text{arcsin}\frac{x}{a}+C$ $a\in R$
$\int \frac{dx}{a^2+x^2} dx=\frac{1}{2}\text{arctan}\frac{x}{a}+C$ $a\in R$
$\int \frac{dx}{a^2-x^2} dx=\frac{1}{2a}\ln \left|\frac{x+a}{x-a}\right|+C$ $a\in R$
Integration Properties
$\int kf(x)dx = k\int f(x)dx$$\int (f(x)\pm g(x))dx = \int f(x)dx \pm \int g(x)dx$
Correct:
Wrong:
Unsolved problems: