Write down the integral of the following (i) `sinx+cos x` (ii)` cosec x (cosec x + cot x)`

Answer

To solve the given integrals step by step, we will break down each part of the question.

### Part (i): Integral of \( \sin x + \cos x \)

1. **Write the integral**:
\[ \int (\sin x + \cos x) \, dx \]

2. **Separate the integral**:
\[ \int \sin x \, dx + \int \cos x \, dx \]

3. **Integrate \( \sin x \)**:
The integral of \( \sin x \) is:
\[ -\cos x + C_1 \]

4. **Integrate \( \cos x \)**:
The integral of \( \cos x \) is:
\[ \sin x + C_2 \]

5. **Combine the results**:
\[ \int (\sin x + \cos x) \, dx = -\cos x + \sin x + C \]
where \( C = C_1 + C_2 \) is the constant of integration.

### Final Answer for Part (i):
\[ \int (\sin x + \cos x) \, dx = -\cos x + \sin x + C \]

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### Part (ii): Integral of \( \csc x (\csc x + \cot x) \)

1. **Write the integral**:
\[ \int \csc x (\csc x + \cot x) \, dx \]

2. **Distribute \( \csc x \)**:
\[ \int (\csc^2 x + \csc x \cot x) \, dx \]

3. **Integrate \( \csc^2 x \)**:
The integral of \( \csc^2 x \) is:
\[ -\cot x + C_3 \]

4. **Integrate \( \csc x \cot x \)**:
The integral of \( \csc x \cot x \) is:
\[ -\csc x + C_4 \]

5. **Combine the results**:
\[ \int \csc x (\csc x + \cot x) \, dx = -\cot x - \csc x + C \]
where \( C = C_3 + C_4 \) is the constant of integration.

### Final Answer for Part (ii):
\[ \int \csc x (\csc x + \cot x) \, dx = -\cot x - \csc x + C \]

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