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EXAMPLE 2Finding the Antiderivatives of a Function

Find all the antiderivatives of $f(x)=e^{x}+\dfrac{6}{x^{2}}-\sin x$ .

Since $f$ is the sum of three functions, we use the Sum Rule. That is, we find the antiderivatives of each function individually and then add. $f(x) =e^{x}+6x^{-2}-\sin x$

Since the antiderivatives of $e^{x}$ are $e^{x}+{C}_{1}$ , the antiderivatives of $\tfrac{6}{x^{2}}$ are $-\tfrac{6}{x}+{C}_{2}$ , and the antiderivatives of $\sin$ $x$ are $-\cos$ $x + {C}_{3}$ , the constant $C$ in Example 2 is actually the sum of the constants $C_{1},$ $C_{2},$ and $C_{3}.$

An antiderivative of $e^{x}$ is $e^{x}.$ An antiderivative of $6x^{-2}$ is $6\cdot \frac{x^{-2+1}}{-2+1}=6\cdot \dfrac{x^{-1}}{-1}=-\dfrac{6}{x}$

Finally, an antiderivative of $\sin x$ is $-\;\cos x$ . Then all the antiderivatives of the function $f$ are given by $F(x) =e^{x}-\frac{6}{x}+\cos x+C$

where $C$ is a constant.