Wiskunde Multiple Choice Vraag

$\int_{1}^{e^2}\frac{\ln x}{\sqrt x}\:dx=$	A. 1
B. 2
C. e
D. 4
E. e²

$\int_{1}^{e^2}\frac{\ln x}{\sqrt x}\:dx=$	A. 1
B. 2
C. e
D. 4
E. e²

$\inline\\\int_{1}^{e^2}\frac{\ln x}{\sqrt x}\:dx=2\int_{1}^{e^2}\!\ln x\;d\sqrt x\;\overset{P.I.}{=}\;2\left(\left[\ln x\cdot\sqrt x\right]_1^{e^2}-\int_{1}^{e^2}\sqrt x\:d\ln x\right)\\=2\left[\sqrt x\cdot\ln x\right]_1^{e^2}-2\int_{1}^{e^2}\frac{x^{\frac12}}{x}dx=2\left[\sqrt x\cdot\ln x\right]_1^{e^2}-2\int_{1}^{e^2}x^{-\frac12}\;dx\\=2\left[\sqrt x\cdot\ln x\right]_1^{e^2}-2\left[\frac{x^{\frac12}}{\frac12}\right]_1^{e^2}=2\left(e.\ln e^2-\sqrt1.\ln 1\right)-4(e^1-1)\\=4e-2.0-4e+4=4$