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Find f'(x) if f(x) = integral_0^4x dt / t^5 + 1.

Question:

Find {eq}f'(x) {/eq} if {eq}\displaystyle f(x) = \int_0^{4x} \frac {dt}{t^5 + 1} {/eq}.

Fundamental Theorems of Calculus

There are two fundamental theorems of calculus. The first fundamental theorem states that:

{eq}\int_{a}^{b} f(x) dx = F(b)-F(a) {/eq}

where F(x) is the antiderivative of f(x). The second fundamental theorem states that given:

{eq}F(x)=\int_{a}^{x} f(t) dt {/eq}

we have the derivative given by {eq}F'(x)=f(x) {/eq}. As with the rules for differentiation, remembering these two theorems involving integrals can greatly simplify integration questions.

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The Fundamental Theorem of Calculus

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Chapter 12 / Lesson 10
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The fundamental theorem of calculus is one of the most important points to understand in mathematics. Learn to define the formula of the fundamental theorem of calculus and explore examples of it put into practice.


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