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Section 5.6 : Definition of the Definite Integral

12. Differentiate the following integral with respect to \(x\).

\[\int_{{3{x^{\,2}}}}^{{ - 1}}{{\frac{{{{\bf{e}}^t} - 1}}{t}dt}}\] Show Solution

This is nothing more than a quick application of the Fundamental Theorem of Calculus, Part I.

Note however, that we’ll need to interchange the limits to get the lower limit to a number and the \(x\)’s in the upper limit as required by the theorem. Also, note that because the upper limit is not just \(x\) we’ll need to use the Chain Rule, with the “inner function” as \(3{x^2}\).

The derivative is,

\[\frac{d}{{dx}}\left[ {\int_{{3{x^{\,2}}}}^{{ - 1}}{{\frac{{{{\bf{e}}^t} - 1}}{t}dt}}} \right] = \frac{d}{{dx}}\left[ { - \int_{{ - 1}}^{{3{x^{\,2}}}}{{\frac{{{{\bf{e}}^t} - 1}}{t}dt}}} \right] = - \left( {6x} \right)\frac{{{{\bf{e}}^{3{x^{\,2}}}} - 1}}{{3{x^2}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{2 - 2{{\bf{e}}^{3{x^{\,2}}}}}}{x}}}\]