`(i) int(sinx)/(sqrt(4+ cos^(2) x)) dx " "(ii) int(x^(2))/(sqrt(9+x^(6)))dx`

Answer

To solve the given integrals step by step, we will tackle each part separately.

### Part (i): \(\int \frac{\sin x}{\sqrt{4 + \cos^2 x}} \, dx\)

**Step 1: Substitution**
Let \( t = \cos x \). Then, the derivative \( dt = -\sin x \, dx \) or \( \sin x \, dx = -dt \).

**Step 2: Change of Variables**
Substituting \( t \) into the integral, we have:
\[ \int \frac{\sin x}{\sqrt{4 + \cos^2 x}} \, dx = \int \frac{-dt}{\sqrt{4 + t^2}} \]

**Step 3: Simplifying the Integral**
This can be rewritten as:
\[ -\int \frac{dt}{\sqrt{4 + t^2}} \]

**Step 4: Recognizing the Integral Form**
The integral \(\int \frac{dt}{\sqrt{a^2 + t^2}}\) has a known solution:
\[ \int \frac{dt}{\sqrt{a^2 + t^2}} = \ln |t + \sqrt{a^2 + t^2}| + C \]
In our case, \( a = 2 \), so:
\[ -\int \frac{dt}{\sqrt{4 + t^2}} = -\ln |t + \sqrt{4 + t^2}| + C \]

**Step 5: Back Substitution**
Now, substituting back \( t = \cos x \):
\[ -\ln |\cos x + \sqrt{4 + \cos^2 x}| + C \]

### Final Answer for Part (i):
\[ \int \frac{\sin x}{\sqrt{4 + \cos^2 x}} \, dx = -\ln |\cos x + \sqrt{4 + \cos^2 x}| + C \]

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### Part (ii): \(\int \frac{x^2}{\sqrt{9 + x^6}} \, dx\)

**Step 1: Substitution**
Let \( u = x^3 \). Then, the derivative \( du = 3x^2 \, dx \) or \( dx = \frac{du}{3x^2} \).

**Step 2: Change of Variables**
Substituting \( u \) into the integral, we have:
\[ \int \frac{x^2}{\sqrt{9 + x^6}} \, dx = \int \frac{x^2}{\sqrt{9 + u^2}} \cdot \frac{du}{3x^2} = \frac{1}{3} \int \frac{1}{\sqrt{9 + u^2}} \, du \]

**Step 3: Recognizing the Integral Form**
The integral \(\int \frac{du}{\sqrt{a^2 + u^2}}\) has a known solution:
\[ \int \frac{du}{\sqrt{a^2 + u^2}} = \ln |u + \sqrt{a^2 + u^2}| + C \]
In our case, \( a = 3 \):
\[ \frac{1}{3} \int \frac{1}{\sqrt{9 + u^2}} \, du = \frac{1}{3} \left( \ln |u + \sqrt{9 + u^2}| + C \right) \]

**Step 4: Back Substitution**
Now, substituting back \( u = x^3 \):
\[ \frac{1}{3} \left( \ln |x^3 + \sqrt{9 + x^6}| + C \right) \]

### Final Answer for Part (ii):
\[ \int \frac{x^2}{\sqrt{9 + x^6}} \, dx = \frac{1}{3} \ln |x^3 + \sqrt{9 + x^6}| + C \]

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