Q. 35 - Calculus

Question

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each the derivatives expressed in Exercises 35–48.

$\frac{d}{dx}{\int }_x^4{e}^{t^2+1}.dt$

Step-by-Step Solution

Answer

The derivative expression of $\frac{d}{dx}{\int }_x^4{e}^{t^2+1}.dt$ is $-e^{x^2+1}$.

Step 1. Given Information.

The derivative:

$\frac{d}{dx}{\int }_x^4{e}^{t^2+1}.dt$

Step 2. Second Fundamental theorem of calculus.

f is continuous on [a,b] for all $x\in [a,b]$, then

$\frac{d}{dx}{\int }_e^{x}f\left(t\right).dt=f\left(x\right)$.

Step 3. Find the derivative expression.

By Second Fundamental theorem of calculus,

$$\begin{gathered} \frac{d}{dx}{\int }_x^4{e}^{t^2+1}.dt=\frac{d}{dx}-{\int }_4^x{e}^{t^2+1}.dt \\ =-e^{x^2+1} \end{gathered}$$