Question

Evaluate the following integral as the limit of a sum:-​

Answer 1

GIVEN :–

$\\ \bf \to \int_{2}^{5} {e}^{3x}.dx$

TO FIND :–

• Value of integration = ?

SOLUTION :–

• Let the function –

$\\ \bf \implies I = \int_{2}^{5} {e}^{3x}.dx$

• Let 3x = t –

• Differentiate with respect to 't' –

$\\ \bf \implies3 \dfrac{dx}{dt} = 1$

$\\ \bf \implies3dx = dt$

$\\ \bf \implies dx = \dfrac{dt}{3}$

• So that –

$\\ \bf \implies I = \dfrac{1}{3} \int_{2}^{5} {e}^{t}.dt$

• We know that –

$\\ \bf \implies \int{e}^{x}.dx = {e}^{x} + c$

$\\ \bf \implies I = \dfrac{1}{3} \bigg[{e}^{t} \bigg]_{2}^{5}$

• Now replace 't' –

$\\ \bf \implies I = \dfrac{1}{3} \bigg[{e}^{3x} \bigg]_{2}^{5}$

$\\ \bf \implies I = \dfrac{1}{3}[{e}^{3(5)} - {e}^{3(2)} ]$

$\\ \bf \implies I = \dfrac{1}{3}({e}^{15} - {e}^{6})$

$\\\large \implies{ \boxed{ \bf I = \dfrac{ {e}^{6}}{3}({e}^{9} -1)}}$