If f:[-1,1] →[-(1)/(2), (1)/(2)], f(x)=(x)/(1+x^2), The area...

If $f:[-1,1] \rightarrow\left[-\frac{1}{2}, \frac{1}{2}\right], f(x)=\frac{x}{1+x^{2}}$, The area bounded by $y=f^{-1}(x), X$ -axis, $x=\frac{1}{2}, x=-\frac{1}{2}$ is (a) $\frac{1}{2} \log e$ (b) $\log \left(\frac{e}{2}\right)$ (c) $\frac{1}{2} \log \frac{e}{3}$ (d) $\frac{1}{2} \log \left(\frac{e}{2}\right)$

If $f:[-1,1] \rightarrow\left[-\frac{1}{2}, \frac{1}{2}\right], f(x)=\frac{x}{1+x^{2}}$, The area bounded
by $y=f^{-1}(x), X$ -axis, $x=\frac{1}{2}, x=-\frac{1}{2}$ is
(a) $\frac{1}{2} \log e$
(b) $\log \left(\frac{e}{2}\right)$
(c) $\frac{1}{2} \log \frac{e}{3}$
(d) $\frac{1}{2} \log \left(\frac{e}{2}\right)$

Key Concepts

Integration Techniques

Integration is a primary method in calculus used to compute areas, volumes, and other quantities. Various techniques, such as substitution or handling variable transformations, are often employed to simplify the integration process, especially when dealing with the inverses of functions or when the region of integration has been modified by a change of variable.

Symmetry in Functions

Symmetry in functions, like even or odd properties, can simplify the evaluation of integrals. When a function or its inverse exhibits symmetry within a specific interval, it often enables reduction of computation by either allowing for the cancellation of symmetric parts or by using symmetric limits, making the overall integration process more straightforward.

Inverse Functions

An inverse function reverses the relationship of the original function, mapping outputs back to their original inputs. In calculus, ensuring the existence of an inverse (which typically requires the function to be injective on the interval of interest) is crucial for problems that involve switching roles between dependent and independent variables, such as when finding the area bounded by the graph of an inverse function.

Area Under a Curve

Computing the area under a curve using definite integrals is a foundational concept in calculus. This technique involves summing an infinite number of infinitesimally thin rectangles to yield the total area between the curve, the x-axis, and specified vertical boundaries. It is particularly useful for calculating the region enclosed by transformed functions, including inverse functions.

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