Question

Evaluate the definite integral.


`\int\limits^7_0 {e^(x)sin(x) } \, dx`

Answer 1

To solve the integral
`\rm\int_0^7 e^x \sin(x) dx\\`, we can use integration by parts. Let u = sin(x) and dv =
`\rm e^x` dx, then we have:

`\begin{align} \rm\int \rm e^x \sin(x) dx &= \rm-e^x \cos(x) + \rm\int e^x \cos(x) dx \\&= \rm -e^x \cos(x) + e^x \sin(x) - \int e^x \sin(x) dx\end{align}`

Rearranging, we get:

`\begin{align}2 \rm \int e^x \sin(x) dx &= \rm e^x (\sin(x) - \cos(x)) \bigg|^7_0 \\& \rm= e^7 (\sin(7) - \cos(7)) - 1\end{align}`

Dividing both sides by 2, we get:

`\rm\int_0^7 e^x \sin(x) dx = (e^7 (\sin(7) - \cos(7)) - 1)/(2) \\`

Therefore, the value of the integral is

`\rm \boxed{ \rm(e^7 (\sin(7) - \cos(7)) - 1)/(2)}`