Question

Evaluate: $\int x{e}^{x^{2}log2}\cdot e^{x^2}dx=$

• A. $\frac{2^{x^2}\cdot e^{x^2}}{2(1+log2)}$
• B. $\frac{2^{x^2}\cdot e^{x^2}}{1+log2}$
• C. $\frac{e^{x^{2}log2}\cdot e^{x^2}}{log2}$
• D. $\frac{(2e{)}^{x^2}}{log\left(2e\right)}$

Answer 1

Answer: (A)

$\frac{2^{x^2}\cdot e^{x^2}}{2(1+log2)}$

$\int x{e}^{x^{2}log2}.e^{x^2}dx=\int x{e}^{(x^{2}log2+x^2)}dx$

$=\int x{e}^{x^2(1+log2)}dx$

assume, $x^2(1+log2)=t⟹2x(1+log2)dx=dt$

substituting in the above equation we get

$\frac{1}{2(1+log2)}\int e^{t}dt=\frac{1}{2(1+log2)}{e}^t+c$

on substituting $t$ we get

$=\frac{e^{x^2(1+log2)}}{2(1+log2)}+c$

$=\frac{e^{x^2}.(e^{log2}{)}^{x^2}}{2(1+log2)}+c$

$=\frac{e^{x^2}{.2}^{x^2}}{2(1+log2)}+c$ (since, $elo{g}_{e}2=2$)