Evaluate: (i) `int1/(a^2-b^2\ x^2)\ dx` (ii) `int1/(a^2\ x^2-b^2)\ dx`

Answer

To evaluate the integrals given in the question, we will tackle each one step by step.

### (i) Evaluate the integral:
\[ \int \frac{1}{a^2 - b^2 x^2} \, dx \]

**Step 1: Factor out the constant**
We can factor out \(b^2\) from the denominator:
\[ \int \frac{1}{a^2 - b^2 x^2} \, dx = \int \frac{1}{b^2 \left(\frac{a^2}{b^2} - x^2\right)} \, dx = \frac{1}{b^2} \int \frac{1}{\frac{a^2}{b^2} - x^2} \, dx \]

**Hint for Step 1:** Factor out constants from the integral to simplify the expression.

**Step 2: Use the integral formula**
We use the formula for the integral:
\[ \int \frac{1}{a^2 - x^2} \, dx = \frac{1}{2a} \log \left| \frac{a + x}{a - x} \right| + C \]
In our case, let \(a = \frac{a}{b}\):
\[ \frac{1}{b^2} \cdot \frac{1}{2 \cdot \frac{a}{b}} \log \left| \frac{\frac{a}{b} + x}{\frac{a}{b} - x} \right| + C \]

**Hint for Step 2:** Recall the integral formula for \(\int \frac{1}{a^2 - x^2} \, dx\) and substitute appropriately.

**Step 3: Simplify the expression**
Now, simplify the expression:
\[ = \frac{1}{2ab} \log \left| \frac{a + bx}{a - bx} \right| + C \]

**Final Answer for (i):**
\[ \int \frac{1}{a^2 - b^2 x^2} \, dx = \frac{1}{2ab} \log \left| \frac{a + bx}{a - bx} \right| + C \]

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### (ii) Evaluate the integral:
\[ \int \frac{1}{a^2 x^2 - b^2} \, dx \]

**Step 1: Factor out the constant**
We can factor out \(a^2\) from the denominator:
\[ \int \frac{1}{a^2 x^2 - b^2} \, dx = \int \frac{1}{a^2 \left(x^2 - \frac{b^2}{a^2}\right)} \, dx = \frac{1}{a^2} \int \frac{1}{x^2 - \frac{b^2}{a^2}} \, dx \]

**Hint for Step 1:** Factor out constants from the integral to simplify the expression.

**Step 2: Use the integral formula**
We use the formula for the integral:
\[ \int \frac{1}{x^2 - a^2} \, dx = \frac{1}{2a} \log \left| \frac{x - a}{x + a} \right| + C \]
In our case, let \(a = \frac{b}{a}\):
\[ = \frac{1}{a^2} \cdot \frac{1}{2 \cdot \frac{b}{a}} \log \left| \frac{x - \frac{b}{a}}{x + \frac{b}{a}} \right| + C \]

**Hint for Step 2:** Recall the integral formula for \(\int \frac{1}{x^2 - a^2} \, dx\) and substitute appropriately.

**Step 3: Simplify the expression**
Now, simplify the expression:
\[ = \frac{a}{2b} \log \left| \frac{ax - b}{ax + b} \right| + C \]

**Final Answer for (ii):**
\[ \int \frac{1}{a^2 x^2 - b^2} \, dx = \frac{a}{2b} \log \left| \frac{ax - b}{ax + b} \right| + C \]

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