Question

Evaluate limit 0to ln3 ∫ e^x(1+e^x)^1\2dx .

Answer 1

$$\displaystyle\int_{0}^{\ln3}e^x(1+e^x)^{{1 \over 2}}dx$$

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

$u=1+e^x, du=e^xdx$

$$\int e^x(1+e^x)^{{1 \over 2}}dx=\int u^{{1 \over 2}}du=\dfrac{2}{3}u^{{3 \over 2}}+C=$$

$$=\dfrac{2}{3}(1+e^x)^{{3 \over 2}}+C=$$

$$\displaystyle\int_{0}^{\ln3}e^x(1+e^x)^{{1 \over 2}}dx=\big[\dfrac{2}{3}(1+e^x)^{{3 \over 2}}\big]\begin{matrix} \ln3 \\ 0 \end{matrix}=$$

$$=\dfrac{2}{3}(1+3)^{{3 \over 2}}-\dfrac{2}{3}(1+1)^{{3 \over 2}}=$$

$$=\dfrac{16}{3}-\dfrac{4\sqrt{2}}{3}$$