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Antiderivative Formula

An Antiderivative is a form of integral without the upper and lower limits. In an antiderivative of a function f, the derivative f is equal to differentiable function F. This process of solving antiderivatives is called as antidifferention. In simple words it is the operation opposite to differention, the process of solving a derivative. An anti-derivative is also mentioned as primitive integral and indefinite integral. And this process is related to definite integrals through the fundamental theorem of calculus.

Here are few basic formulas of antiderivatives:

$\int e^{x}dx=e^{x}+c\\
\int a^{x}dx=\frac{a^{x}}{In a}+c\\
\int \frac{1}{x}dx=In\left | x \right |+c\\
\int sin\ x\ dx = -cos\ x\ + c\\
\int cos\ x\ dx = sin\ x + c\\
\int sec^{2}\ x\ dx = tan\ x\ +\ c\\
\int csc^{2}\ x\ dx = -cot\ x\ +\ c\\
\int sec\ x\ tan\ x\ dx\ = sec\ x\ + c\\
\int \frac{1}{1 + x^{2}} dx = arctan x + c\\
\int \frac{1}{\sqrt{1 - x^{2}}} dx = arcsin x + c\\
\int csc\ x\ cot\ x\ dx = - csc\ x\ +\ c\\
\int sec\ x\ dx = -ln\left | sec\ x + tan\ x \right | + c\\
\int csc \ x\ dx = -ln\left | csc\ x + cot\ x \right | + c\\
\int x^{n}dx = \frac{x^{n+1}}{n + 1} + c, when\ n\neq 1$

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Antiderivative Problems

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Below are the problems on Antiderivative with solutions :

Solved Examples

Question 1: Calculate the anti-derivative of the function: f(x) = 3x2 + x ?

Given function: f(x) = 3x2 + x
Anti-derivative of the given function: $\int$ f(x) = $\int$ ( 3x2 + x)

$\int$ ( 3x2 + x) = $\int$ ( 3x2) + $\int$ (x) = 3 $\frac{x^{3}}{3}$ + $\frac{x^{2}}{2}$ (Since $\int$ (x)n = $\frac{x^{(n + 1)}}{n + 1}$ + c)

$\Rightarrow$ x3 + $\frac{x^{2}}{2}$ + c

Question 2: Find the anti-derivative of the function: f(x) = $\cos (3x)$ ?

Given function: f(x) = $\cos (3x)$

Anti-derivative of the given function: $\int$ f(x) = $\cos (2x)$

$\int$ $\cos (2x)$ = $\frac{\sin\ 3x}{3}$  = (Since $\int$ $\cos (ax)$  = $\frac{\sin (ax)}{a}$ + c)

$\Rightarrow$ $\frac{\sin\ 3x}{3}$ + c

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