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Integrating Standard Functions

\(\DeclareMathOperator{cosec}{cosec}\)

Integrating standard functions
  • \[\int x^n \, dx =\frac{x^{n+1}}{n+1}+c\]
  • \[\int e^x \, dx =e^x+c\]
  • \[\int \frac{1}{x} \, dx =\ln |x|+c\]
  • \[\int \cos x \, dx =\sin x+c\]
  • \[\int \sin x \, dx =-\cos x+c\]
  • \[\int sec^2x \, dx =\tan x+c\]
  • \[\int \cosec x \cot x \, dx =-\cosec x +c\]
  • \[\int \cosec^2x \, dx =-\cot x+c\]
  • \[\int \sec x \tan x\, dx =\sec x +c\]
\(\int \left( 2\cos x+\frac{3}{x}-\sqrt{ x } \right) \, dx\)
  • \(=2\sin x+3\ln |x|-\frac{2}{3}x^{\frac{3}{2}}+c\)
\(\int 3\sec^2x+\frac{5}{x}+\frac{2}{x^2} \, dx\)
  • \(3\tan x+5\ln |x|-2x^{-1}+c\)
  • \(3\tan x+5\ln |x|-\frac{2}{x}+c\)
\(\int 5e^x-4\sin x+2x^3 \, dx\)
  • \(5e^x+4\cos x+\frac{x^4}{2}+c\)
\(\int 2(\sin x-\cos x+x) \, dx\)
  • \(2\left( -\cos x-\sin x+\frac{x^2}{2} \right)+c\)
\(\int 3\sec x\tan x-\frac{2}{x} \, dx\)
  • \(3\sec x-2\ln |x|+c\)
\(\int \frac{1}{2x}+2\cosec^2x \, dx\)
  • \(\frac{1}{2}\ln |x|-2\cot x+c\)
\(\int \frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3} \, dx\)
  • \(\ln |x|-\frac{1}{x}-\frac{2}{x^2}+c\)
\(\int e^x+\sin x+\cos x \, dx\)
  • \(e^x-\cos x+\sin x\)
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