Find {eq}f {/eq} if {eq}f'(x) = \dfrac{4x^2+7}{1+x^2} {/eq} and {eq}f(0) = -1 {/eq}.

Question:

Find {eq}f {/eq} if {eq}f'(x) = \dfrac{4x^2+7}{1+x^2} {/eq} and {eq}f(0) = -1 {/eq}.

Indefinite Integral:

One of the uses of indefinite integrals is in obtaining general and specific antiderivatives.

For specific antiderivatives, we need initial values.

The indefinite integral that we'll use is:

{eq}\displaystyle \int\frac{\mathrm{d}x}{1+x^2}=\tan^{-1}(x) + C {/eq}

Answer and Explanation:

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Let's get {eq}f(x) {/eq} utilizing {eq}f(x) = \displaystyle \int \dfrac{4x^2+7}{1+x^2} \, \mathrm{d}x {/eq}.

To be able to use {eq}\displaystyle...

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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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