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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,

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