The Fundamental Theorem of Calculus to find the derivative of the function. y = ?^ 5_ 4 ? 3 x u^...

Question:

Use the Fundamental Theorem of Calculus to find the derivative of the function {eq}\displaystyle y(x)=\int_{4-3x}^{5}\frac{u^{3}}{1+u^{2}} \, du \, . {/eq}

The Fundamental Theorem of Calculus:

The fundamental theorem of calculus connects the two basic operations of calculus: differentiation and integration. It has two parts:

1) The first fundamental theorem of calculus says that, 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}

2) The second fundamental theorem of calculus says that, if {eq}f {/eq} is a function on the interval {eq}[a, b] {/eq}, {eq}c {/eq} is a point in the interval {eq}[a, b] {/eq}, and we define

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

then {eq}F'(x)=f(x) {/eq}.

Let {eq}F(u) {/eq} be an antiderivative of the function {eq}\frac{u^3}{1+u^2} {/eq}. Then, by the first fundamental theorem of calculus, we have:

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