Question

Calculando a integral obtêm-se:

Answer 1

$\displaystyle\int\limits_{0}^{(1/4)}3e^{4y}dy$

Resolvemos facilmente a integral fazendo substituição de variáveis

Chamando 4y de u, temos

$limites~de~int.~\begin{cases}u_{0}=4\cdot0=0\\u_{1}=4\cdot(\frac{1}{4})=1\end{cases}$

Derivando u:

$du=4y^{1-1}dy~~~\therefore~~~du=4dy~~~\therefore~~~dy=(\frac{1}{4})du$

Então, substituindo 4y por u e os limites de integração encontrados em u:

$\displaystyle\int\limits_{0}^{(1/4)}3e^{4y}dy=\int\limits_{0}^{1}3e^{u}\left(\frac{1}{4}\right)du\\\\\\\int\limits_{0}^{(1/4)}3e^{4y}dy=\int\limits_{0}^{1}\left(\dfrac{3}{4}\right)e^{u}du$

Podemos retirar constantes da integral:

$\displaystyle\int\limits_{0}^{(1/4)}3e^{4y}dy=\dfrac{3}{4}\int\limits_{0}^{1}e^{u}du\\\\\\\int\limits_{0}^{(1/4)}3e^{4y}dy=\dfrac{3}{4}[e^{u}]_{0}^{1}\\\\\\\int\limits_{0}^{(1/4)}3e^{4y}dy=\dfrac{3}{4}(e^{1}-e^{0})\\\\\\\boxed{\boxed{\int\limits_{0}^{(1/4)}3e^{4y}dy=\dfrac{3}{4}(e-1)}}$