`int(1-x)sqrt(1+x)dx`

Answer

To solve the integral \( \int (1-x) \sqrt{1+x} \, dx \), we will follow these steps:

### Step 1: Substitution
Let \( t = 1 + x \). Then, the differential \( dx \) becomes \( dt \). We also need to express \( 1 - x \) in terms of \( t \):
\[ 1 - x = 1 - (t - 1) = 2 - t \]
Now, substitute these into the integral:
\[ \int (1-x) \sqrt{1+x} \, dx = \int (2 - t) \sqrt{t} \, dt \]

### Step 2: Expand the Integral
Now we can expand the integrand:
\[ \int (2 - t) \sqrt{t} \, dt = \int (2\sqrt{t} - t\sqrt{t}) \, dt = \int (2t^{1/2} - t^{3/2}) \, dt \]

### Step 3: Integrate Each Term
Now, we can integrate each term separately:
\[ \int 2t^{1/2} \, dt - \int t^{3/2} \, dt \]
Using the power rule for integration:
\[ \int t^{n} \, dt = \frac{t^{n+1}}{n+1} + C \]
we get:
\[ = 2 \cdot \frac{t^{3/2}}{3/2} - \frac{t^{5/2}}{5/2} = \frac{4}{3} t^{3/2} - \frac{2}{5} t^{5/2} + C \]

### Step 4: Substitute Back
Now we substitute back \( t = 1 + x \):
\[ = \frac{4}{3} (1+x)^{3/2} - \frac{2}{5} (1+x)^{5/2} + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int (1-x) \sqrt{1+x} \, dx = \frac{4}{3} (1+x)^{3/2} - \frac{2}{5} (1+x)^{5/2} + C \]

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