Solve for: (d)/(dx) (integral from 1 to x^2 of ln(5+t^2) t)

Expression: $\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \int_{ 1 }^{ {x}^{2} } \ln\left({5+{t}^{2}}\right) \mathrm{d} t \right)$

Substitute $u$ for ${x}^{2}$

$\frac{ \mathrm{d} }{ \mathrm{d}x} \left( \int_{ 1 }^{ u } \ln\left({5+{t}^{2}}\right) \mathrm{d} t \right)$

Use the chain rule to find the derivative

$\frac{ \mathrm{d} }{ \mathrm{d}u} \left( \int_{ 1 }^{ u } \ln\left({5+{t}^{2}}\right) \mathrm{d} t \right) \times \frac{ \mathrm{d}^u }{ \mathrm{d}^x }$

To find the derivative, apply the second fundamental theorem of calculus

$\ln\left({5+{u}^{2}}\right) \times \frac{ \mathrm{d}^u }{ \mathrm{d}^x }$

Rewrite the derivative

$\ln\left({5+{u}^{2}}\right) \times \frac{ \mathrm{d} }{ \mathrm{d}x} \left( u \right)$

Substitute back $u={x}^{2}$

$\ln\left({5+{\left( {x}^{2} \right)}^{2}}\right) \times \frac{ \mathrm{d} }{ \mathrm{d}x} \left( {x}^{2} \right)$

Find the derivative

$\ln\left({5+{\left( {x}^{2} \right)}^{2}}\right) \times 2x$

Simplify the expression

$2\ln\left({5+{x}^{4}}\right) \times x$