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 each of the integrals step by step.

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

**Step 1:** Identify the integral formula.
The integral of \( x^n \) is given by:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( n \neq -1 \).

**Step 2:** Apply the formula.
Here, \( n = -\frac{3}{2} \). Thus, we have:
\[ n + 1 = -\frac{3}{2} + 1 = -\frac{1}{2} \]
So,
\[ \int x^{-\frac{3}{2}} \, dx = \frac{x^{-\frac{1}{2}}}{-\frac{1}{2}} + C = -2 x^{-\frac{1}{2}} + C \]

**Final Answer for (1):**
\[ \int x^{-\frac{3}{2}} \, dx = -2 x^{-\frac{1}{2}} + C \]

---

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

**Step 1:** Identify the value of \( \sin 60^\circ \).
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]

**Step 2:** Factor out the constant.
\[ \int \sin 60^\circ \, dx = \int \frac{\sqrt{3}}{2} \, dx = \frac{\sqrt{3}}{2} \int dx \]

**Step 3:** Integrate \( dx \).
\[ \int dx = x + C \]

**Final Answer for (2):**
\[ \int \sin 60^\circ \, dx = \frac{\sqrt{3}}{2} x + C \]

---

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

**Step 1:** Factor out the constant.
\[ \int \frac{1}{10x} \, dx = \frac{1}{10} \int \frac{1}{x} \, dx \]

**Step 2:** Integrate \( \frac{1}{x} \).
\[ \int \frac{1}{x} \, dx = \ln |x| + C \]

**Final Answer for (3):**
\[ \int \frac{1}{10x} \, dx = \frac{1}{10} \ln |x| + C \]

---

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

**Step 1:** Integrate each term separately.
\[ \int (2x^3 - x^2 + 1) \, dx = \int 2x^3 \, dx - \int x^2 \, dx + \int 1 \, dx \]

**Step 2:** Apply the integral formula to each term.
1. For \( \int 2x^3 \, dx \):
\[ = 2 \cdot \frac{x^{4}}{4} = \frac{1}{2} x^4 \]
2. For \( \int x^2 \, dx \):
\[ = \frac{x^{3}}{3} \]
3. For \( \int 1 \, dx \):
\[ = x \]

**Step 3:** Combine the results.
\[ \int (2x^3 - x^2 + 1) \, dx = \frac{1}{2} x^4 - \frac{1}{3} x^3 + x + C \]

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

---

### Summary of Answers:
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. \( \int (2x^3 - x^2 + 1) \, dx = \frac{1}{2} x^4 - \frac{1}{3} x^3 + x + C \)

---