Write down the integral of the following (i) `x^(2//3)+1` (ii) `(3/sqrtx+5x^4)`

Answer

To solve the given integrals, we will apply the basic rules of integration.

### Part (i): Integral of \( x^{\frac{2}{3}} + 1 \)

1. **Identify the integral**:
\[ \int (x^{\frac{2}{3}} + 1) \, dx \]

2. **Separate the integral**:
\[ \int x^{\frac{2}{3}} \, dx + \int 1 \, dx \]

3. **Apply the power rule of integration**:
The power rule states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(where \( n \neq -1 \))} \]
- For \( x^{\frac{2}{3}} \):
\[ n = \frac{2}{3} \implies \int x^{\frac{2}{3}} \, dx = \frac{x^{\frac{2}{3} + 1}}{\frac{2}{3} + 1} = \frac{x^{\frac{5}{3}}}{\frac{5}{3}} = \frac{3}{5} x^{\frac{5}{3}} \]
- For \( 1 \):
\[ \int 1 \, dx = x \]

4. **Combine the results**:
\[ \int (x^{\frac{2}{3}} + 1) \, dx = \frac{3}{5} x^{\frac{5}{3}} + x + C \]

### Final Answer for Part (i):
\[ \int (x^{\frac{2}{3}} + 1) \, dx = \frac{3}{5} x^{\frac{5}{3}} + x + C \]

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### Part (ii): Integral of \( \frac{3}{\sqrt{x}} + 5x^4 \)

1. **Identify the integral**:
\[ \int \left( \frac{3}{\sqrt{x}} + 5x^4 \right) \, dx \]

2. **Rewrite the expression**:
\[ \frac{3}{\sqrt{x}} = 3x^{-\frac{1}{2}} \implies \int (3x^{-\frac{1}{2}} + 5x^4) \, dx \]

3. **Separate the integral**:
\[ \int 3x^{-\frac{1}{2}} \, dx + \int 5x^4 \, dx \]

4. **Apply the power rule of integration**:
- For \( 3x^{-\frac{1}{2}} \):
\[ \int 3x^{-\frac{1}{2}} \, dx = 3 \cdot \frac{x^{-\frac{1}{2} + 1}}{-\frac{1}{2} + 1} = 3 \cdot \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 6\sqrt{x} \]
- For \( 5x^4 \):
\[ \int 5x^4 \, dx = 5 \cdot \frac{x^{4 + 1}}{4 + 1} = 5 \cdot \frac{x^5}{5} = x^5 \]

5. **Combine the results**:
\[ \int \left( \frac{3}{\sqrt{x}} + 5x^4 \right) \, dx = 6\sqrt{x} + x^5 + C \]

### Final Answer for Part (ii):
\[ \int \left( \frac{3}{\sqrt{x}} + 5x^4 \right) \, dx = 6\sqrt{x} + x^5 + C \]

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