Question

The number of distinct solutions of ${\int }_0^x(4t^2+1)e^{2t^2}dt=5x$ for $x\ge 0$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

Let $g\left(x\right)=\underset{0}{\overset{x}{\int }}(4t^2+1)e^{2t^2}dt$
$2t^2=u$ or $t=\frac{\sqrt{u}}{\sqrt{2}}$
$g\left(x\right)=\underset{0}{\overset{2x^2}{\int }}(2u+1)e^u.\frac{1}{2\sqrt{2}\sqrt{u}}du$
$=\frac{1}{\sqrt{2}}{\int }_0^{2x^2}(\sqrt{u}+\frac{1}{2\sqrt{u}})e^{x}dx=\frac{1}{\sqrt{2}}[\sqrt{u}.e^u{]}_0^{2x^2}$
$x.e^{2x^2}=5x\Rightarrow x=0$ or $x=+\sqrt{\frac{\ln 5}{2}}$