Calculate the integral: integral from 0 to (pi/2) e^x sin

Calculate the integral: $\int_0^{\pi/2} e^x \sin x \, dx$.

We will use integration by parts twice to evaluate this integral.

Step 1: First integration by parts

Let $u = \sin x$, so $u' = \cos x$; and $dv = e^x \, dx$, so $v = e^x$.

\[\int_0^{\pi/2} e^x \sin x \, dx = \left[e^x \sin x\right]_0^{\pi/2} - \int_0^{\pi/2} e^x \cos x \, dx\]

Step 2: Evaluate the boundary term

\[\left[e^x \sin x\right]_0^{\pi/2} = e^{\pi/2} \cdot 1 - 1 \cdot 0 = e^{\pi/2}\]

So:

\[\int_0^{\pi/2} e^x \sin x \, dx = e^{\pi/2} - \int_0^{\pi/2} e^x \cos x \, dx\]

Step 3: Second integration by parts on $\int_0^{\pi/2} e^x \cos x \, dx$

Let $u = \cos x$, so $u' = -\sin x$; and $dv = e^x \, dx$, so $v = e^x$.

\[\int_0^{\pi/2} e^x \cos x \, dx = \left[e^x \cos x\right]_0^{\pi/2} + \int_0^{\pi/2} e^x \sin x \, dx\]

Step 4: Evaluate the boundary term

\[\left[e^x \cos x\right]_0^{\pi/2} = e^{\pi/2} \cdot 0 - 1 \cdot 1 = -1\]

So:

\[\int_0^{\pi/2} e^x \cos x \, dx = -1 + \int_0^{\pi/2} e^x \sin x \, dx\]

Step 5: Substitute back and solve

Let $I = \int_0^{\pi/2} e^x \sin x \, dx$. From Step 2: $I = e^{\pi/2} - \int_0^{\pi/2} e^x \cos x \, dx$. From Step 4: $\int_0^{\pi/2} e^x \cos x \, dx = -1 + I$. Substituting:

\[I = e^{\pi/2} - (-1 + I) = e^{\pi/2} + 1 - I \quad\Rightarrow\quad 2I = e^{\pi/2} + 1 \quad\Rightarrow\quad I = \frac{e^{\pi/2} + 1}{2}\]

Numerical verification:

Using $e^{\pi/2} \approx 4.8105$: $\frac{e^{\pi/2} + 1}{2} \approx 2.9052$.

Alternative approach - general formula:

For $\int e^{ax} \sin(bx) \, dx$, integration by parts twice gives $\frac{e^{ax}(a\sin(bx) - b\cos(bx))}{a^2 + b^2} + C$. For $a=b=1$:

\[\int e^x \sin x \, dx = \frac{e^x(\sin x - \cos x)}{2} + C\]

Using this:

\[\int_0^{\pi/2} e^x \sin x \, dx = \left[\frac{e^x(\sin x - \cos x)}{2}\right]_0^{\pi/2} = \frac{e^{\pi/2}}{2} + \frac{1}{2} = \frac{e^{\pi/2} + 1}{2}\]

This confirms our result!

Final answer:

\[\boxed{\int_0^{\pi/2} e^x \sin x \, dx = \frac{e^{\pi/2} + 1}{2}}\]

Or approximately:

\[\boxed{\int_0^{\pi/2} e^x \sin x \, dx \approx 2.905}\]