Q. 36 - Calculus

Question

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each the derivatives expressed in Exercises 35–48.
$\frac{d}{dx}{\int }_2^x\frac{\sin t}{t}.dt$

Step-by-Step Solution

Answer

The derivative expression of $\frac{d}{dx}{\int }_2^x\frac{\sin t}{t}.dt$ is $\frac{\sin x}{x}$.

Step 1. Given Information.

The derivative: 

$\frac{d}{dx}{\int }_2^x\frac{\sin t}{t}.dt$

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, 

$\frac{d}{dx}{\int }_2^x\frac{\sin t}{t}.dt=\frac{\sin x}{x}$