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Use the Second Fundamental Theorem of Calculus to find F'(x). F(x) = \int^x _{-3} \sqrt {t^4 + 1} dt

Question:

Use the Second Fundamental Theorem of Calculus to find {eq}\displaystyle F'(x) {/eq}.

{eq}\displaystyle F(x) = \int^x _{-3} \sqrt {t^4 + 1} dt {/eq}

The Fundamental Theorem of Calculus:

The fundamental theorem of calculus has two parts:

1. If {eq}F {/eq} is an antiderivative of {eq}f {/eq} on the interval {eq}[a, b] {/eq}, then

{eq}\displaystyle \int_a^b f(x) \, dx = F(b)-F(a) \, . {/eq}

This is the first fundamental theorem of calculus.

2. Suppose {eq}f {/eq} is a function on {eq}[a, b] {/eq}. Define another function {eq}F {/eq} on {eq}[a, b] {/eq} by

{eq}\displaystyle F(x)=\int_c^x f(t) \, dt {/eq}

for some {eq}c {/eq} in {eq}[a, b] {/eq}. Then {eq}F'(x)=f(x) {/eq}. This is the second fundamental theorem of calculus.

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The Fundamental Theorem of Calculus

from Math 104: Calculus

Chapter 12 / Lesson 10
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