Question

Ajuda!!ALGUÉM???
Calcule o valor de cada integral definida
a)∫ xsenxdx x=0 e x= x/2

b) ∫ lnxdx x=1 e x=e

Answer 1

a)

$\large\boxed{\int xsenx\ dx}\\\\ u = x\rightarrow\ du=dx\\\\\ dv=senx\ dx\rightarrow\ v=-cosx$

$\int\ xsenx\ dx = -xcosx(-\int\ -cosx\ dx)\\\\ \int\ xsenx\ dx = -xcosx+senx\\\\ \large\boxed{\int\ xsenx\ dx=senx-xcosx+C}$

Agora, pelo teorema

$\int\limits^2_0 {xsenx} \, dx = (sen2-2(cos2))-(sen0-0cos0)\\\\ \large\boxed{\int\limits^2_0 {xsenx} \, dx =sen2-2cos2}$

Esse valor dá aproximadamente 1,74....

b)

$\large\boxed{\int\ lnx\ dx}\\\\ u=lnx\rightarrow\ du=\frac{1}{x}\ dx\\\\ dv=dx\rightarrow\ v=x\\ \\ \int\ lnx\ dx=x*lnx-\int\ x*\frac{1}{x}\ dx\\\\ \large\boxed{\int\ lnx\ dx=xlnx-x+C}$

Aplicando o teorema

$\int\limits^e_1 {lnx} \, dx =(e*lne-e)-(1*ln1-1)\\\\ \int\limits^e_1 {lnx} \, dx=(e*1-e)-(0-1)\\\\ \large\boxed{ \int\limits^e_1 {lnx} \, dx=1}$