Evaluate the integrals.
(i) `int 2x^(7) ` dx

Answer

To evaluate the integral \( \int 2x^{7} \, dx \), we can follow these steps:

### Step 1: Identify the integral
We start with the integral:
\[ \int 2x^{7} \, dx \]

### Step 2: Use the constant multiple rule
According to the constant multiple rule of integration, we can factor out the constant:
\[ \int 2x^{7} \, dx = 2 \int x^{7} \, dx \]

### Step 3: Apply the power rule of integration
Next, we apply the power rule of integration, which states that:
\[ \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \]
For our case, \( n = 7 \):
\[ \int x^{7} \, dx = \frac{x^{7+1}}{7+1} + C = \frac{x^{8}}{8} + C \]

### Step 4: Substitute back into the integral
Now we substitute this result back into our expression:
\[ 2 \int x^{7} \, dx = 2 \left( \frac{x^{8}}{8} + C \right) \]

### Step 5: Simplify the expression
Distributing the 2 gives us:
\[ = \frac{2x^{8}}{8} + 2C = \frac{x^{8}}{4} + C' \]
where \( C' = 2C \) is still a constant of integration.

### Final Answer
Thus, the evaluated integral is:
\[ \int 2x^{7} \, dx = \frac{x^{8}}{4} + C \]

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