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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}

Answer and Explanation: 1

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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...

See full answer below.


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Indefinite Integrals as Anti Derivatives

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Chapter 12 / Lesson 11
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Indefinite integrals, an integral of the integrand which does not have upper or lower limits, can be used to identify individual points at specific times. Learn more about the fundamental theorem, use of antiderivatives, and indefinite integrals through examples in this lesson.


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