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Find {eq}f(x) {/eq} given {eq}\displaystyle f''(x) = 2 + \cos (x), f(9) = -1, f(\frac {\pi}{2}) = 0 {/eq}.

Question:

Find {eq}f(x) {/eq} given {eq}\displaystyle f''(x) = 2 + \cos (x), f(9) = -1, f(\frac {\pi}{2}) = 0 {/eq}.

Indefinite Integral of a Real-Valued Function:

Let {eq}\displaystyle f {/eq} be a real-valued function of {eq}\displaystyle x {/eq}. If there exists a function {eq}\displaystyle F {/eq} such that {eq}\displaystyle f\left( x \right) = F'\left( x \right),\,\,\,\forall \,\,\,x {/eq} then {eq}\displaystyle F {/eq} is said to be an antiderivative of {eq}\displaystyle f {/eq}. The general form, i.e. {eq}\displaystyle F(x)+C {/eq}, where {eq}\displaystyle C {/eq} is any integration constant (independent of {eq}\displaystyle x {/eq}), is called an indefinite integral of {eq}\displaystyle f {/eq} and it is usually denoted by {eq}\displaystyle \int_{}^{} {} f\left( x \right)dx = F\left( x \right) + C {/eq}.

Answer and Explanation: 1

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Given that {eq}\displaystyle f''(x) = 2 + \cos (x) {/eq}.

Integrating, we get

{eq}\displaystyle \begin{align} & \int_{}^{} {f''\left( x...

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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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