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Section B.2 Integration Formulas
List B.2.1. Basic Rules
\(\displaystyle \int c\cdot f(x)\,dx=c\int f(x)\,dx\)
\(\displaystyle \int \bigl(f(x)\pm g(x)\bigr)\,dx = \int f(x)\, dx \pm \int g(x)\, dx\)
\(\displaystyle \int 0\,dx = C\)
\(\displaystyle \int 1\,dx=x+C\)
\(\displaystyle \int e^x\,dx=e^x+C\)
\(\displaystyle \int \ln(x)\,dx=x\ln(x) -x +C\)
\(\displaystyle \int a^x\,dx=\frac{1}{\ln(a)}\cdot a^x+C\)
\(\displaystyle \int \frac{1}{x}\,dx =\ln \abs{x} + C\)
\(\displaystyle \int x^n\,dx=\frac{1}{n+1}x^{n+1}+C, n\neq -1\)
\(\displaystyle \int \cos(x)\,dx=\sin(x)+C\)
\(\displaystyle \int \sin(x)\,dx=-\cos(x)+C\)
\(\displaystyle \int \tan(x)\,dx=-\ln \abs{\cos(x)}+C\)
\(\displaystyle \int \sec(x)\,dx=\ln \abs{\sec(x)+\tan(x)}+C\)
\(\displaystyle \int \csc(x)\,dx=-\ln \abs{\csc(x)+\cot(x)}+C\)
\(\displaystyle \int \cot(x)\,dx=\ln \abs{\sin(x)}+C\)
\(\displaystyle \int \sec^2(x)\,dx=\tan(x)+C\)
\(\displaystyle \int \csc^2(x)\,dx=-\cot(x)+C\)
\(\displaystyle \int \sec(x)\tan(x)\,dx=\sec(x)+C\)
\(\displaystyle \int \csc(x)\cot(x)\,dx=-\csc(x)+C\)
\(\displaystyle \int \cos^2(x)\,dx=\frac12x+\frac14\sin\big(2x\big)+C\)
\(\displaystyle \int \sin^2(x)\,dx=\frac12x-\frac14\sin\big(2x\big)+C\)
\(\displaystyle \int \frac{1}{x^2+a^2}\,dx = \frac1a\tan^{-1}\left(\frac xa\right)+C\)
\(\displaystyle \int \frac{1}{\sqrt{a^2-x^2}} = \sin^{-1}\left(\frac xa\right)+C\)
\(\displaystyle \int \frac{1}{x\sqrt{x^2-a^2}} = \frac1a\sec^{-1}\left(\frac{\abs{x}}{a}\right)+C\)
\(\displaystyle \int \cosh(x)\,dx=\sinh(x)+C\)
\(\displaystyle \int \sinh(x)\,dx=\cosh(x)+C\)
\(\displaystyle \int \tanh(x)\,dx=\ln(\cosh(x))+C\)
\(\displaystyle \int \coth(x)\,dx=\ln \abs{\sinh(x)}+C\)
\(\displaystyle \int \frac{1}{\sqrt{x^2-a^2}}\, dx =\ln\abs{x+\sqrt{x^2-a^2}}+C\)
\(\displaystyle \int \frac{1}{\sqrt{x^2+a^2}}\, dx=\ln\abs{x+\sqrt{x^2+a^2}}+C\)
\(\displaystyle \int \frac{1}{a^2-x^2}\, dx =\frac{1}{2a}\ln\abs{\frac{a+x}{a-x}}+C\)
\(\displaystyle \int \frac{1}{x\sqrt{a^2-x^2}}\, dx = \frac{1}{a}\ln\left(\frac{x}{a+\sqrt{a^2-x^2}}\right)+C\)
\(\displaystyle \int \frac{1}{x\sqrt{x^2+a^2}}\, = \frac{1}{a}\ln\abs{\frac{x}{a+\sqrt{x^2+a^2}}}+C\)
