`int_0^1 (x e^x +sin((pix)/4)) dx`

Answer

To solve the integral \( I = \int_0^1 \left( x e^x + \sin\left(\frac{\pi x}{4}\right) \right) dx \), we can break it into two parts:

\[ I = \int_0^1 x e^x \, dx + \int_0^1 \sin\left(\frac{\pi x}{4}\right) \, dx \]

### Step 1: Solve \( \int_0^1 x e^x \, dx \)

We will use integration by parts for this integral. Let:
- \( u = x \) (then \( du = dx \))
- \( dv = e^x \, dx \) (then \( v = e^x \))

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):

\[ \int_0^1 x e^x \, dx = \left[ x e^x \right]_0^1 - \int_0^1 e^x \, dx \]

Calculating the boundary term:

\[ \left[ x e^x \right]_0^1 = 1 \cdot e^1 - 0 \cdot e^0 = e - 0 = e \]

Now calculate the second integral:

\[ \int_0^1 e^x \, dx = \left[ e^x \right]_0^1 = e - 1 \]

Putting it all together:

\[ \int_0^1 x e^x \, dx = e - (e - 1) = 1 \]

### Step 2: Solve \( \int_0^1 \sin\left(\frac{\pi x}{4}\right) \, dx \)

We will use substitution for this integral. Let:
- \( u = \frac{\pi x}{4} \) (then \( du = \frac{\pi}{4} dx \) or \( dx = \frac{4}{\pi} du \))

Change the limits:
- When \( x = 0 \), \( u = 0 \)
- When \( x = 1 \), \( u = \frac{\pi}{4} \)

Now we can rewrite the integral:

\[ \int_0^1 \sin\left(\frac{\pi x}{4}\right) \, dx = \int_0^{\frac{\pi}{4}} \sin(u) \cdot \frac{4}{\pi} \, du = \frac{4}{\pi} \int_0^{\frac{\pi}{4}} \sin(u) \, du \]

Calculating the integral:

\[ \int \sin(u) \, du = -\cos(u) \]

Thus:

\[ \int_0^{\frac{\pi}{4}} \sin(u) \, du = \left[-\cos(u)\right]_0^{\frac{\pi}{4}} = -\cos\left(\frac{\pi}{4}\right) + \cos(0) = -\frac{1}{\sqrt{2}} + 1 = 1 - \frac{1}{\sqrt{2}} \]

Putting it all together:

\[ \int_0^1 \sin\left(\frac{\pi x}{4}\right) \, dx = \frac{4}{\pi} \left(1 - \frac{1}{\sqrt{2}}\right) \]

### Final Calculation

Now combine both parts:

\[ I = \int_0^1 x e^x \, dx + \int_0^1 \sin\left(\frac{\pi x}{4}\right) \, dx = 1 + \frac{4}{\pi} \left(1 - \frac{1}{\sqrt{2}}\right) \]

### Final Result

Thus, the final result for the integral is:

\[ I = 1 + \frac{4}{\pi} \left(1 - \frac{1}{\sqrt{2}}\right) \]