009B Sample Midterm 3, Problem 2

State the fundamental theorem of calculus, and use this theorem to find the derivative of

$F(x)=\int _{\cos(x)}^{5}{\frac {1}{1+u^{10}}}~du.$

Foundations:
What does Part 1 of the Fundamental Theorem of Calculus say is the derivative of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G(x)=\int _{x}^{5}{\frac {1}{1+u^{10}}}~du?}
First, we need to switch the bounds of integration.
So, we have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G(x)=-\int _{5}^{x}{\frac {1}{1+u^{10}}}~du.}
By Part 1 of the Fundamental Theorem of Calculus, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G'(x)=-{\frac {1}{1+x^{10}}}.}

Solution:

Step 1:
The Fundamental Theorem of Calculus has two parts.
The Fundamental Theorem of Calculus, Part 1
Let $f$ be continuous on $[a,b]$ and let $F(x)=\int _{a}^{x}f(t)~dt.$
Then, $F$ is a differentiable function on $(a,b)$ and $F'(x)=f(x).$
The Fundamental Theorem of Calculus, Part 2
Let $f$ be continuous on $[a,b]$ and let $F$ be any antiderivative of $f.$
Then, $\int _{a}^{b}f(x)~dx=F(b)-F(a).$

Step 2:
First, we have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F(x)=-\int _{5}^{\cos(x)}{\frac {1}{1+u^{10}}}~du.}
Now, let $g(x)=\cos(x)$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G(x)=\int _{5}^{x}{\frac {1}{1+u^{10}}}~du.}
So, $F(x)=-G(g(x)).$
Hence, $F'(x)=-G'(g(x))g'(x)$ by the Chain Rule.

Step 3:
Now, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g'(x)=-\sin(x).}
By the Fundamental Theorem of Calculus, $G'(x)={\frac {1}{1+x^{10}}}.$
Hence, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle F'(x)=-{\frac {1}{1+\cos ^{10}x}}(-\sin(x))={\frac {\sin(x)}{1+\cos ^{10}x}}.}

Final Answer:
The Fundamental Theorem of Calculus, Part 1
Let $f$ be continuous on $[a,b]$ and let $F(x)=\int _{a}^{x}f(t)~dt.$
Then, $F$ is a differentiable function on $(a,b)$ and $F'(x)=f(x).$
The Fundamental Theorem of Calculus, Part 2
Let $f$ be continuous on $[a,b]$ and let $F$ be any antiderivative of $f.$
Then, $\int _{a}^{b}f(x)~dx=F(b)-F(a).$
$F'(x)={\frac {\sin(x)}{1+\cos ^{10}x}}$

Return to Sample Exam