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,
\[\frac{d}{{dx}}\left[ {\int_{7}^{{\sin \left( {6x} \right)}}{{\sqrt {{t^2} + 4} dt}}} \right] = \require{bbox} \bbox[2pt,border:1px solid black]{{6\cos \left( {6x} \right)\sqrt {{{\sin }^2}\left( {6x} \right) + 4} }}\]