Question

question 3 (1 point)
evaluate ( int_{0}^{sqrt{ln (5)}} x e^{x^{2}} d x ).
( e^{sqrt{ln (5)}}-1 )
( \frac{1}{2}left(e^{sqrt{ln (5)}}-1
ight) )
2
( \frac{1}{5} )
0

Explanation:

Step1: Substitute \( u = x^2 \)

\( du = 2x dx \implies \frac{1}{2}du = x dx \)

Step2: Change limits of integration

When \( x=0 \), \( u=0^2=0 \); when \( x=\sqrt{\ln 5} \), \( u=(\sqrt{\ln 5})^2=\ln 5 \)

Step3: Rewrite integral with \( u \)

\( \int_{0}^{\ln 5} e^u \cdot \frac{1}{2}du = \frac{1}{2}\int_{0}^{\ln 5} e^u du \)

Step4: Integrate \( e^u \)

\( \frac{1}{2}[e^u]_{0}^{\ln 5} = \frac{1}{2}(e^{\ln 5} - e^0) \)

Step5: Simplify using \( e^{\ln a}=a \)

\( \frac{1}{2}(5 - 1) = 2 \)

Answer:

2