Question

If \[ \int\frac{(1-x^{2})\,dx}{\sqrt{x}\sqrt{(1+x^{2})^{3}}} =\alpha\frac{x^{\beta}}{(1+x^{2})^{\gamma}}+C, \] \(\alpha,\beta,\gamma\in\mathbb{R}\) and \(C\) is constant of integration, then \(\alpha:\beta:\gamma\) will be:

• A. $4:1:1$
• B. $2:2:\frac{1}{2}$
• C. $\frac{1}{6}:2:\frac{1}{2}$
• D. $1:2:\frac{1}{2}$

Hint:

Whenever you see an expression like $(1-x^2)$ in the numerator and $(1+x^2)$ in the denominator, try dividing the top and bottom by $x$ or $x^2$. This is a reliable way to convert the expression into matching reciprocal groups of the form $(x \pm 1/x)$ that simplify nicely.

Answer 1

The Correct Option is A

Concept: Algebraic integrals that feature combinations of $x$ and $(1+x^2)$ can be solved by factoring out high powers of $x$ to create terms matching the derivative of $\left(x + \frac{1}{x}\right)$ or similar reciprocal groups. Step 1: Rearrange and factor the integrand expression.
Let us rewrite the given integral by factoring out $x$ from inside the denominator bracket to see the reciprocal structure clearly: \[ I = \int \frac{1-x^2}{\sqrt{x} \cdot (1+x^2)^{\frac{3}{2}}}\,dx \] Let us divide the numerator and denominator parameters by $x^2$: \[ I = \int \frac{\frac{1}{x^2} - 1}{\sqrt{x} \cdot \frac{(1+x^2)^{\frac{3}{2}}}{x^2}}\,dx = \int \frac{\frac{1}{x^2} - 1}{\sqrt{x} \cdot x^{-\frac{1}{2}} \cdot \left(x + \frac{1}{x}\right)^{\frac{3}{2}}}\,dx \] Notice that $\sqrt{x} \cdot x^{-\frac{1}{2}} = 1$, which gives: \[ I = \int \frac{\frac{1}{x^2} - 1}{\left(x + \frac{1}{x}\right)^{\frac{3}{2}}}\,dx \quad \cdots (1) \]

Step 2: Apply a variable substitution step.
Notice that the numerator is closely related to the derivative of the term inside the denominator bracket. Let us substitute: \[ u = x + \frac{1}{x} \quad \Rightarrow \quad du = \left(1 - \frac{1}{x^2}\right)dx = -\left(\frac{1}{x^2} - 1\right)dx \] Substitute $u$ and $du$ directly back into our equation (1): \[ I = \int \frac{-du}{u^{\frac{3}{2}}} = -\int u^{-\frac{3}{2}}\,du \]

Step 3: Integrate and convert back to the variable $x$.
Integrate using the standard power integration rule: \[ I = -\left( \frac{u^{-\frac{1}{2}}}{-\frac{1}{2}} \right) + C = \frac{2}{\sqrt{u}} + C \] Substitute our original definition of $u = x + \frac{1}{x} = \frac{x^2+1}{x}$ back into the equation: \[ I = \frac{2}{\sqrt{\frac{x^2+1}{x}}} + C = 2 \cdot \frac{\sqrt{x}}{\sqrt{1+x^2}} + C = 2 \cdot \frac{x^{\frac{1}{2}}}{(1+x^2)^{\frac{1}{2}}} + C \]

Step 4: Extract the coefficients to find the ratio.
Comparing this equation with the template form $\alpha\frac{x^{\beta}}{(1+x^{2})^{\gamma}}$: \[ \alpha = 2, \quad \beta = \frac{1}{2}, \quad \gamma = \frac{1}{2} \] We are asked to find the ratio of these three real numbers: \[ \alpha : \beta : \gamma = 2 : \frac{1}{2} : \frac{1}{2} \] Multiply all terms in the ratio by 2 to clear the fractions and find the matching integer ratio: \[ \text{Ratio} = (2 \times 2) : \left(\frac{1}{2} \times 2\right) : \left(\frac{1}{2} \times 2\right) = 4 : 1 : 1 \] This matches option (A) perfectly.