# Evaluate the following integrals. int x^210^{x^3}dx

Question
Integrals
Evaluate the following integrals.
$$\int x^210^{x^3}dx$$

2021-02-25
$$\text{The given integral is,}$$
$$\int x^210^{x^3}dx$$
$$\text{Using substitution method of integration, on substituting }x^3\text{ with t}$$
$$x^3=t$$
$$3x^2dx=dt$$
$$x^2dx=\frac{dt}{3}$$
$$\text{Then the transformed integral we get is,}$$
$$I=\int\frac{10^t}{3}dt$$
$$=\frac{1}{3}\int 10^tdt$$
$$\text{Using exponential rule, }\int a^xdx=\frac{a^x}{\ln(a)}+C$$
$$I=\frac{1}{3}\int10^tdt$$
$$=\frac{1}{3}[\frac{10^t}{\ln(10)}]+C$$
$$\text{On puttinf back the value of t, we get}$$
$$I=\frac{10^{x^3}}{3\ln(10)}+C$$
$$\text{Therefore, the value of the given integral is }\frac{10^{x^3}}{3\ln(10)}+C$$\)

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