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Antiderivative Formula
The antiderivative formula is used to get back the original function, for which a derivative had been performed. In general usage, the antiderivative formula refers to the process of integration. Integration returns it back to the original function, for which a derivative had been performed. The antiderivative is the reverse of the derivative process. Antiderivative formula applies to algebraic expressions, logarithmic functions, trigonometric functions, and inverse trigonometric functions.
What is the Antiderivative Formula?
The antiderivative for the function f'(x) gives back the original function f(x). Further, the function is derived to get back the original function.
\[\int f'(x).dx = f(x) + C\]
Some of the additional formulas which would be useful for the integration(antiderivative) of a function are as follows.
 \(\int x^n.dx = \dfrac{x^{n + 1}}{n + 1} + C \)
 \(\int 1.dx = x + C \)
 \(\int e^x.dx = e^x + C \)
 \(\int\frac{1}{x}.dx = Logx + C \)
 \(\int a^x.dx = \dfrac{a^x}{loga} + C \)
Let us solve a few examples to clearly understand the antiderivative of a function.
Solved Examples on Antiderivative Formula

Example 1: Find the antiderivative of 16x^{7} using antiderivative formula.
Solution:
\(\begin{align}\int 16x^7.dx &=16.\frac{x^8}{8} \\&=2x^8 + C\end{align}\)
Answer: \(\int 16x^7.dx =2x^8 + C \) 
Example 2: Simplify and find the value of \( \int Cosecx(Cosecx  Cotx).dx \).
Solution:
\(\begin{align} \int Cosecx(Cosecx  Cotx).dx &= \int (Cosec^2x  Cotx.Cosecx).dx \\&=\int Cosec^2x.dx  \int Cotx.Cosecx.dx \\&= Cotx  (Cosecx) \\&= Cotx + Cosecx\\& = Cosecx  Cotx + C \end{align}\)
Answer: \(\int Cosecx(Cosecx  Cotx).dx = Cosecx  Cotx + C \)
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