Question

Write a value of

\[\int e^{2 x^2 + \ln x} \text{ dx}\]

Answer 1

\[\text{ Let I }= \int \left( e^{2 x^2 + \ln x} \right)dx\]
\[ = \int \left( e^{2 x^2} \times e^{ \text{ ln   x}} \right)dx\]
\[ = \int e^{2 x^2} . x \text{ dx}\]
\[\text{ Let 2}\ x^2 = t\]
\[ \Rightarrow \text{ 4x dx} = dt\]
\[ \Rightarrow\text{  x dx} = \frac{dt}{4}\]
\[ \therefore I = \frac{1}{4}\int e^t dt\]
\[ = \frac{1}{4} e^t + C\]
\[ = \frac{1}{4} e^{2 x^2} + C \left( \because t = 2 x^2 \right)\]