## Calculus: Early Transcendentals (2nd Edition)

$$\frac{{2x}}{{{x^4} + 4}}$$
\eqalign{ & \frac{d}{{dx}}\int_0^{{x^2}} {\frac{{dt}}{{{t^2} + 4}}} \cr & {\text{using The Fundamental Theorem }}\left( {{\text{part 1}}} \right) \cr & A\left( x \right) = \int_a^x {f\left( t \right)dt{\text{ }} \to {\text{ }}} A'\left( x \right) = \frac{d}{{dx}}\int_a^x {f\left( t \right)dt} = f\left( x \right) \cr & {\text{the upper limit is }}{x^2},{\text{ so is needed to apply the chain rule}} \cr & \frac{d}{{dx}}\int_0^{{x^2}} {\frac{{dt}}{{{t^2} + 4}}} = \frac{{dy}}{{dx}} = \frac{{dy}}{{du}}\frac{{du}}{{dx}} \cr & {\text{with }}u = {x^2} \cr & = \left( {\frac{d}{{du}}\int_0^u {\frac{{dp}}{{{p^2}}}} } \right)\frac{d}{{dx}}\left( {{x^2}} \right) \cr & = \left( {\frac{d}{{du}}\int_0^u {\frac{{dt}}{{{t^2} + 4}}} } \right)\left( {2x} \right) \cr & {\text{by the fundamental theorem }}\left( {{\text{part 1}}} \right) \cr & = \left( {\frac{1}{{{u^2} + 4}}} \right)\left( {2x} \right) \cr & = \left( {\frac{1}{{{{\left( {{x^2}} \right)}^2} + 4}}} \right)\left( {2x} \right) \cr & = \frac{{2x}}{{{x^4} + 4}} \cr}