\(\text{The given integral is,}\)

\(\int x^210^{x^3}dx\)

\(\text{Using substitution method of integration, on substituting }x^3\text{ with t}\)

\(x^3=t\)

\(3x^2dx=dt\)

\(x^2dx=\frac{dt}{3}\)

\(\text{Then the transformed integral we get is,}\)

\(I=\int\frac{10^t}{3}dt\)

\(=\frac{1}{3}\int 10^tdt\)

\(\text{Using exponential rule, }\int a^xdx=\frac{a^x}{\ln(a)}+C\)

\(I=\frac{1}{3}\int10^tdt\)

\(=\frac{1}{3}[\frac{10^t}{\ln(10)}]+C\)

\(\text{On puttinf back the value of t, we get}\)

\(I=\frac{10^{x^3}}{3\ln(10)}+C\)

\(\text{Therefore, the value of the given integral is }\frac{10^{x^3}}{3\ln(10)}+C\)\)

\(\int x^210^{x^3}dx\)

\(\text{Using substitution method of integration, on substituting }x^3\text{ with t}\)

\(x^3=t\)

\(3x^2dx=dt\)

\(x^2dx=\frac{dt}{3}\)

\(\text{Then the transformed integral we get is,}\)

\(I=\int\frac{10^t}{3}dt\)

\(=\frac{1}{3}\int 10^tdt\)

\(\text{Using exponential rule, }\int a^xdx=\frac{a^x}{\ln(a)}+C\)

\(I=\frac{1}{3}\int10^tdt\)

\(=\frac{1}{3}[\frac{10^t}{\ln(10)}]+C\)

\(\text{On puttinf back the value of t, we get}\)

\(I=\frac{10^{x^3}}{3\ln(10)}+C\)

\(\text{Therefore, the value of the given integral is }\frac{10^{x^3}}{3\ln(10)}+C\)\)