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Questions: Algebra BusinessCalculus

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Calculus
Integrals
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Find the derivative of \(\displaystyle F(x) = \int\limits_{1}^{\sin{\left(x \right)}} \ln{\left(t \right)}\, dt\).

Using the Fundamental Theorem of Calculus part I and the chain rule with \(\displaystyle u=\sin{\left(x \right)}\) and \(\displaystyle F(u)=\int\limits_{1}^{u} \ln{\left(t \right)}\, dt\) gives: \(\displaystyle F'(x)=\frac{dF}{du}\frac{du}{dx} = \left(\ln{\left(u \right)}\right)\left(\cos{\left(x \right)}\right)=\log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}\)

Download \(\LaTeX\) 

\begin{question}Find the derivative of $F(x) = \int\limits_{1}^{\sin{\left(x \right)}} \ln{\left(t \right)}\, dt$. 
    \soln{9cm}{Using the Fundamental Theorem of Calculus part I and the chain rule with $u=\sin{\left(x \right)}$  and $F(u)=\int\limits_{1}^{u} \ln{\left(t \right)}\, dt$ gives: $F'(x)=\frac{dF}{du}\frac{du}{dx} = \left(\ln{\left(u \right)}\right)\left(\cos{\left(x \right)}\right)=\log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the derivative of  <img class="equation_image" title=" \displaystyle F(x) = \int\limits_{1}^{\sin{\left(x \right)}} \ln{\left(t \right)}\, dt " src="/equation_images/%20%5Cdisplaystyle%20F%28x%29%20%3D%20%5Cint%5Climits_%7B1%7D%5E%7B%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%7D%20%5Cln%7B%5Cleft%28t%20%5Cright%29%7D%5C%2C%20dt%20" alt="LaTeX:  \displaystyle F(x) = \int\limits_{1}^{\sin{\left(x \right)}} \ln{\left(t \right)}\, dt " data-equation-content=" \displaystyle F(x) = \int\limits_{1}^{\sin{\left(x \right)}} \ln{\left(t \right)}\, dt " /> . </p> </p>
HTML for Canvas
<p> <p>Using the Fundamental Theorem of Calculus part I and the chain rule with  <img class="equation_image" title=" \displaystyle u=\sin{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20u%3D%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle u=\sin{\left(x \right)} " data-equation-content=" \displaystyle u=\sin{\left(x \right)} " />   and  <img class="equation_image" title=" \displaystyle F(u)=\int\limits_{1}^{u} \ln{\left(t \right)}\, dt " src="/equation_images/%20%5Cdisplaystyle%20F%28u%29%3D%5Cint%5Climits_%7B1%7D%5E%7Bu%7D%20%5Cln%7B%5Cleft%28t%20%5Cright%29%7D%5C%2C%20dt%20" alt="LaTeX:  \displaystyle F(u)=\int\limits_{1}^{u} \ln{\left(t \right)}\, dt " data-equation-content=" \displaystyle F(u)=\int\limits_{1}^{u} \ln{\left(t \right)}\, dt " />  gives:  <img class="equation_image" title=" \displaystyle F'(x)=\frac{dF}{du}\frac{du}{dx} = \left(\ln{\left(u \right)}\right)\left(\cos{\left(x \right)}\right)=\log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} " src="/equation_images/%20%5Cdisplaystyle%20F%27%28x%29%3D%5Cfrac%7BdF%7D%7Bdu%7D%5Cfrac%7Bdu%7D%7Bdx%7D%20%3D%20%5Cleft%28%5Cln%7B%5Cleft%28u%20%5Cright%29%7D%5Cright%29%5Cleft%28%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%5Cright%29%3D%5Clog%7B%5Cleft%28%5Csin%7B%5Cleft%28x%20%5Cright%29%7D%20%5Cright%29%7D%20%5Ccos%7B%5Cleft%28x%20%5Cright%29%7D%20" alt="LaTeX:  \displaystyle F'(x)=\frac{dF}{du}\frac{du}{dx} = \left(\ln{\left(u \right)}\right)\left(\cos{\left(x \right)}\right)=\log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} " data-equation-content=" \displaystyle F'(x)=\frac{dF}{du}\frac{du}{dx} = \left(\ln{\left(u \right)}\right)\left(\cos{\left(x \right)}\right)=\log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} " /> </p> </p>