Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 5.6 : Definition of the Definite Integral
11. Differentiate the following integral with respect to \(x\).
\[\int_{7}^{{\sin \left( {6x} \right)}}{{\sqrt {{t^2} + 4} dt}}\] Show SolutionThis is nothing more than a quick application of the Fundamental Theorem of Calculus, Part I.
Note however, that because the upper limit is not just \(x\) we’ll need to use the Chain Rule, with the “inner function” as \(\sin \left( {6x} \right)\).
The derivative is,
\[\frac{d}{{dx}}\left[ {\int_{7}^{{\sin \left( {6x} \right)}}{{\sqrt {{t^2} + 4} dt}}} \right] = \require{bbox} \bbox[2pt,border:1px solid black]{{6\cos \left( {6x} \right)\sqrt {{{\sin }^2}\left( {6x} \right) + 4} }}\]