# Find {eq}\displaystyle{ \rm f } {/eq} that satisfies {eq}\displaystyle{ \rm f'''(x) = 2^x. } {/eq}

## Question:

Find {eq}\displaystyle{ \rm f } {/eq} that satisfies {eq}\displaystyle{ \rm f'''(x) = 2^x. } {/eq}

## Indefinite Integration:

• An indefinite integration of a function gives the primitive function. The function we integrate is nothing but a derivative of its primitive function. We have to add an integration constant to generalize the solution. A constant of integration need to be introduced because when we differentiate a constant term, it will become zero and because we know that we must get the integrand function when we differentiate the integrated function.
• If we integrate a function twice, we need to introduce another integration constant, and similarly whenever we integrate a function, an integration constant needs to be introduced.
• The following integral function will be applied:

{eq}\begin{align} \hspace{1cm}\int a^x\, dx&=\frac{a^x}{\ln a} +C\\[0.3cm] \hspace{1cm} \int x^n\, dx&=\frac { x^{n+1}} {n+1}+C & \left[\text{ This is the power rule of integration } \right]\\[0.3cm] \end{align} {/eq}

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We have the following given data

{eq}\begin{align} f'''(x) &= 2^x \\[0.3cm] f(x) &= ??\\[0.3cm] \end{align} {/eq}

Solution

We have to integrate...