Integrate the following
(1)`intx^(-3/2)dx`
(2)`intsin60^(@)dx`
(3)`int(1)/(10x)dx`
(4)`int(2x^(3)-x^(2)+1)dx`

Answer

Let's solve the integrals step by step.

### Step 1: Integrate \( \int x^{-\frac{3}{2}} \, dx \)

To integrate \( x^{-\frac{3}{2}} \), we use the power rule for integration:

\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

Here, \( n = -\frac{3}{2} \).

Calculating \( n + 1 \):

\[ n + 1 = -\frac{3}{2} + 1 = -\frac{3}{2} + \frac{2}{2} = -\frac{1}{2} \]

Now, applying the power rule:

\[ \int x^{-\frac{3}{2}} \, dx = \frac{x^{-\frac{1}{2}}}{-\frac{1}{2}} + C = -2 x^{-\frac{1}{2}} + C \]

Thus, the result is:

\[ \int x^{-\frac{3}{2}} \, dx = -2 x^{-\frac{1}{2}} + C \]

### Step 2: Integrate \( \int \sin 60^\circ \, dx \)

Since \( \sin 60^\circ = \frac{\sqrt{3}}{2} \), we can treat this as a constant:

\[ \int \sin 60^\circ \, dx = \int \frac{\sqrt{3}}{2} \, dx \]

This simplifies to:

\[ \frac{\sqrt{3}}{2} \int dx = \frac{\sqrt{3}}{2} x + C \]

Thus, the result is:

\[ \int \sin 60^\circ \, dx = \frac{\sqrt{3}}{2} x + C \]

### Step 3: Integrate \( \int \frac{1}{10x} \, dx \)

We can factor out the constant \( \frac{1}{10} \):

\[ \int \frac{1}{10x} \, dx = \frac{1}{10} \int \frac{1}{x} \, dx \]

The integral of \( \frac{1}{x} \) is \( \ln |x| \):

\[ \int \frac{1}{10x} \, dx = \frac{1}{10} \ln |x| + C \]

Thus, the result is:

\[ \int \frac{1}{10x} \, dx = \frac{1}{10} \ln |x| + C \]

### Step 4: Integrate \( \int (2x^3 - x^2 + 1) \, dx \)

We can integrate each term separately:

\[ \int (2x^3 - x^2 + 1) \, dx = \int 2x^3 \, dx - \int x^2 \, dx + \int 1 \, dx \]

Calculating each integral:

1. \( \int 2x^3 \, dx = 2 \cdot \frac{x^4}{4} = \frac{1}{2} x^4 \)
2. \( \int x^2 \, dx = \frac{x^3}{3} \)
3. \( \int 1 \, dx = x \)

Combining these results:

\[ \int (2x^3 - x^2 + 1) \, dx = \frac{1}{2} x^4 - \frac{1}{3} x^3 + x + C \]

Thus, the result is:

\[ \int (2x^3 - x^2 + 1) \, dx = \frac{1}{2} x^4 - \frac{1}{3} x^3 + x + C \]

### Final Results

1. \( \int x^{-\frac{3}{2}} \, dx = -2 x^{-\frac{1}{2}} + C \)
2. \( \int \sin 60^\circ \, dx = \frac{\sqrt{3}}{2} x + C \)
3. \( \int \frac{1}{10x} \, dx = \frac{1}{10} \ln |x| + C \)
4. \( \int (2x^3 - x^2 + 1) \, dx = \frac{1}{2} x^4 - \frac{1}{3} x^3 + x + C \)