If y(x-y)^2=x, then ∫(d x)/(x-3 y)= (A) (1)/(2) ln|(x-y)^2+1|+C (B) (1)/(2) ln|(x-y)^2-1|+C (C)

step 1 So in the question, it is given that y into x minus y whole square equal x. So put, put x minus

If y(x-y)^2=x, then ∫(d x)/(x-3 y)= (A) (1)/(2)...

If $y(x-y)^{2}=x$, then $\int \frac{d x}{x-3 y}=$
(A) $\frac{1}{2} \ln \left|(x-y)^{2}+1\right|+C$
(B) $\frac{1}{2} \ln \left|(x-y)^{2}-1\right|+C$
(C) $\frac{1}{4} \ln \left|(x-y)^{2}+1\right|+C$
(D) none of these.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used when a function is given in a form where the dependent variable is not isolated. In such cases, the derivative with respect to the independent variable is obtained by differentiating both sides of the equation with respect to that variable, while treating the dependent variable as an implicit function. This approach is essential when the relationship between the variables is complex, as it allows determination of the derivative even when an explicit expression is not available.

Substitution Method in Integration

The substitution method, often referred to as u?substitution, is a powerful tool in integration that simplifies the integrand by introducing a new variable. By selecting a substitution that captures a significant part of the integrand or its derivative, the integration process can be transformed into a more straightforward form. This method is widely used in scenarios where the original variable does not lend itself to easy integration, and it is instrumental in reducing integrals to standard forms.

Algebraic Manipulation in Implicit Functions

Algebraic manipulation involves re-arranging, factoring, or substituting parts of an equation to simplify the relationship between variables. When dealing with implicit functions, this strategy can help isolate a composite function or identify patterns that can be matched with standard integral forms or substitutions, thereby bridging the gap between the implicit relationship and the methods of differentiation or integration.

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