Fundamental Calculus

\(y = \int f(x) dx = F(x) + C\)

Note

\(f\): integrand

\(dx\): variable of integration

\(F(x)\): an antiderivative of \(f(x)\)

\(C\): constant of integration

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\(\int x^3\ dx = \frac{x^4}{4} + C\)

\(\boxed{\int kf(x)dx = k \int f(x)dx}\)
\(\boxed{\int[f(x) \pm g(x)]} = \int f(x) dx \pm \int g(x) dx\)
\(\boxed{\int x^n dx = \frac1{n+1} x^{n+1} + C}\) \(\iff\) \(\boxed{\int \frac1x dx = \ln |x| + C}\)
\(\boxed{\int a^x dx = (\frac1{\ln a}) a^x + C}\)
\(\int e^x dx = e^x + C\)
\(\boxed{\int \sin x dx = - \cos x + C}\)
\(\boxed{\cos x dx = \sin x + C}\)
\(\boxed{\int \sec^2 x\ dx = -\tan x + C}\)
\(\int \csc^2 x\ dx = -\cot x + C\)
\(\int \sec x \tan x \ dx = \sec x + C\)
\(\int \csc x \cot x \ dx = - \csc x + C\)
\(\int \dfrac{dx}{\sqrt{1-x^2}} = \arcsin x + C\)
\(\boxed{\int \dfrac{dx}{1+x^2}} = \arctan x + C\)