Paso a paso para resolver con el métodode integración por partes?

<img src="https://tex.z-dn.net/?f=%5Cint%20_2%5E5%5Cfrac%7Bx%5Cln%20%5Cleft%28x%5Cright%29%7D%7B5%7Ddx" />

Se calcula la integral indefinida :

<img src="https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B5%7D%5Cint%20%5C%3Ax%5Cln%20%5Cleft%28x%5Cright%29dx" />

Se aplica la integración por partes :

<img src="https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B5%7D%5Cleft%28%5Cfrac%7B1%7D%7B2%7Dx%5E2%5Cln%20%5Cleft%28x%5Cright%29-%5Cint%20%5Cfrac%7Bx%7D%7B2%7Ddx%5Cright%29" />

Sacamos la constante, se aplical relga de la potencia y se simplifica :

<img src="https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B5%7D%5Cleft%28%5Cfrac%7B1%7D%7B2%7Dx%5E2%5Cln%20%5Cleft%28x%5Cright%29-%5Cfrac%7Bx%5E2%7D%7B4%7D%5Cright%29%2BC" />

Se calculan los límites :

<img src="https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A2%2B%7D%5Cleft%28%5Cfrac%7B1%7D%7B5%7D%5Cleft%28%5Cfrac%7B1%7D%7B2%7Dx%5E2%5Cln%20%5Cleft%28x%5Cright%29-%5Cfrac%7Bx%5E2%7D%7B4%7D%5Cright%29%5Cright%29" />

y

<img src="https://tex.z-dn.net/?f=%5Clim%20_%7Bx%5Cto%20%5C%3A5-%7D%5Cleft%28%5Cfrac%7B1%7D%7B5%7D%5Cleft%28%5Cfrac%7B1%7D%7B2%7Dx%5E2%5Cln%20%5Cleft%28x%5Cright%29-%5Cfrac%7Bx%5E2%7D%7B4%7D%5Cright%29%5Cright%29" />

Por lo que

<img src="https://tex.z-dn.net/?f=%3D%5Cfrac%7B5%7D%7B4%7D%5Cleft%28%5Cln%20%5Cleft%2825%5Cright%29-1%5Cright%29-%5Cfrac%7B1%7D%7B5%7D%5Cleft%28%5Cln%20%5Cleft%284%5Cright%29-1%5Cright%29" />

Se simplifica :

<img src="https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B20%7D%5Cleft%28-21-8%5Cln%20%5Cleft%282%5Cright%29%2B50%5Cln%20%5Cleft%285%5Cright%29%5Cright%29" />

<img src="https://tex.z-dn.net/?f=%5Cleft%28%5Cmathrm%7BDecimal%3A%5Cquad%20%7D%5C%3A2.69634%5Cright%29" />.