The value of `int((ax^2-b)dx)/(xsqrt(c^2x^2-(ax^2+b)^2))` is equal to

Options

A. `(1)/(c)sin^(-1)(ax+(b)/(x))+k`
B. `c sin^(-1)(a+(b)/(x))+k`
C. `sin^(-1)((ax+(b)/(x))/(c))+k`
D. none of these

Explanation

To solve the integral

\[ I = \int \frac{ax^2 - b}{x \sqrt{c^2 x^2 - (ax^2 + b)^2}} \, dx, \]

we can follow these steps:

### Step 1: Simplify the Integral
We can rewrite the integral as:

\[ I = \int \frac{ax^2 - b}{x \sqrt{c^2 x^2 - (ax^2 + b)^2}} \, dx. \]

### Step 2: Factor Out \(x^2\)
Notice that we can factor out \(x^2\) from the square root in the denominator:

\[ I = \int \frac{ax^2 - b}{x \sqrt{x^2(c^2 - \frac{(ax^2 + b)^2}{x^2})}} \, dx. \]

This simplifies to:

\[ I = \int \frac{ax^2 - b}{x^2 \sqrt{c^2 - \left(\frac{(ax^2 + b)}{x}\right)^2}} \, dx. \]

### Step 3: Substitute
Let \(t = ax^2 + b\). Then, differentiating gives us:

\[ dt = 2ax \, dx \quad \Rightarrow \quad dx = \frac{dt}{2ax}. \]

Substituting this into the integral gives:

\[ I = \int \frac{(a - \frac{b}{x^2})}{\sqrt{c^2 - t^2}} \cdot \frac{dt}{2a}. \]

### Step 4: Change of Variables
Now, we can express the integral in terms of \(t\):

\[ I = \frac{1}{2a} \int \frac{(a - \frac{b}{x^2})}{\sqrt{c^2 - t^2}} \, dt. \]

### Step 5: Solve the Integral
We know that:

\[ \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right) + C. \]

Thus, applying this to our integral:

\[ I = \frac{1}{2a} \sin^{-1}\left(\frac{t}{c}\right) + C. \]

### Step 6: Substitute Back
Now, substituting back for \(t\):

\[ I = \frac{1}{2a} \sin^{-1}\left(\frac{ax^2 + b}{c}\right) + C. \]

### Final Answer
Thus, the value of the integral is:

\[ I = \frac{1}{2a} \sin^{-1}\left(\frac{ax^2 + b}{c}\right) + C. \]

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