Anything that is the opposite of a function and has been differentiated in trigonometric terms is known as an anti-derivative. Both the antiderivative and the differentiated function are continuous on a specified interval. In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. Some of the formulas are mentioned below.
Basic Antiderivatives
\(\begin{array}{l}\large If\: f(x) = a,\: then \: F(x) = ax+C\end{array} \)
\(\begin{array}{l}\large If\: f(x) = x^{a},\: then \: F(x) = \frac{x^{a+1}}{a+1} + C\:(unless\:a=-1)\end{array} \)
\(\begin{array}{l}\large If\: f(x) = \frac{1}{x},\: then \: F(x) = \ln (x)+C\end{array} \)
\(\begin{array}{l}\large If\: f(x) = e^{x},\:then\:F(x) = e^{x}+C\end{array} \)
\(\begin{array}{l}\large If\: f(x) = \cos(x),\:then\:F(x) = \sin(x)+C\end{array} \)
\(\begin{array}{l}\large If\: f(x) = \sin(x),\:then\:F(x) = -\cos(x)+C\end{array} \)
\(\begin{array}{l}\large If\: f(x) = \sec^{2}(x),\:then\:F(x) = \tan(x)+C\end{array} \)
These can be written using integral as given below:
\(\begin{array}{l}\large \int e^{x}dx=e^{x}+C\end{array} \)
\(\begin{array}{l}\large \int a^{x}dx=\frac{a^{x}}{\ln a}+C\end{array} \)
\(\begin{array}{l}\large \int \frac{1}{x}dx=\ln\left | x \right |+C\end{array} \)
\(\begin{array}{l}\large \int \cos x\:dx= \sin x +C\end{array} \)
\(\begin{array}{l}\large \int \sec ^{2}x\:dx= \tan x +C\end{array} \)
\(\begin{array}{l}\large \int \sin x\:dx= -\cos x +C\end{array} \)
\(\begin{array}{l}\large \int \csc^{2} x\:dx= -\cot x +C\end{array} \)
\(\begin{array}{l}\large \int \sec x\:\tan x\:dx= \sec x +C\end{array} \)
\(\begin{array}{l}\large \int \frac{1}{1+x^{2}}\:dx= \arctan x +C\end{array} \)
\(\begin{array}{l}\large \int \frac{1}{\sqrt{1-x^{2}}}\:dx= \arcsin x +C\end{array} \)
\(\begin{array}{l}\large \int \csc x \cot x\:dx= -\csc x +C\end{array} \)
\(\begin{array}{l}\large \int\sec x\:dx= \ln \left | \sec x+\tan x \right | +C\end{array} \)
\(\begin{array}{l}\large \int\csc x\:dx= \ln \left |\csc x-\cot x \right | +C\end{array} \)
\(\begin{array}{l}\large \int x^{n}\:dx= \frac{x^{n+1}}{n+1}+C,\: when\:n\neq -1\end{array} \)
\(\begin{array}{l}\large \int \sinh x \:dx= \cosh x+C\end{array} \)
\(\begin{array}{l}\large \int \cosh x \:dx= \sinh x+C\end{array} \)
Antiderivative Rules
\(\begin{array}{l}\large If\:the\:antiderivative\:of\:f(x)\:is\:F(x), and\:the\:antiderivative\:of\:g(x)\:is\:G(x),\:then\:\end{array} \)
1) \(\begin{array}{l}\large The\:Antiderivative\:of\:af(x)+bg(x)\:is\:aF(x)+bG(x)\:(for\:any\:a,b)\end{array} \)
2) \(\begin{array}{l}\large The\:Antiderivative\:of\:f(ax+b)\:is\:\frac{1}{a}F(ax+b)\end{array} \)
\(\begin{array}{l}Therefore,\end{array} \)
\(\begin{array}{l}\large If\:f(x)=(dx+b)^{a}\:then\:F(x)=\frac{1}{d}\frac{(dx+b)^{a+1}}{a+1} +C\:(unless\:a=-1)\end{array} \)
\(\begin{array}{l}\large If\:f(x)=\frac{1}{ax+b}\:then\:F(x)=\frac{1}{a}\ln (ax+b)+C\end{array} \)
\(\begin{array}{l}\large If\:f(x)=e^{ax+b}\:then\:F(x)=\frac{1}{a}e^{ax+b}+C\end{array} \)
\(\begin{array}{l}\large If\:f(x)=\cos(ax+b)\:then\:F(x)=\frac{1}{a}\sin(ax+b)+C\end{array} \)
\(\begin{array}{l}\large If\:f(x)=\sin(ax+b)\:then\:F(x)=-\frac{1}{a}\cos(ax+b)+C\end{array} \)
\(\begin{array}{l}\large If\:f(x)=\sec^{2}(ax+b)\:then\:F(x)=\frac{1}{a}\tan(ax+b)+C\end{array} \)
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