The area of the region between the curves...

The area of the region between the curves $y=\sqrt{\frac{1+\sin x}{\cos x}}$ and $y=\sqrt{\frac{1-\sin x}{\cos x}}$ and bounded by the lines $x=0$ and $x=\frac{\pi}{4}$ is $\quad$ [Single Correct Option 2008] (a) $\int_{0}^{\sqrt{2}-1} \frac{t}{\left(1+t^{2}\right) \sqrt{1-t^{2}}} d t$ (b) $\int_{0}^{\sqrt{2}-1} \frac{4 t}{\left(1+t^{2}\right) \sqrt{1-t^{2}}} d t$ (c) $\int_{0}^{\sqrt{2}+1} \frac{4 t}{\left(1+t^{2}\right) \sqrt{1-t^{2}}} d t$ (d) $\int_{0}^{\sqrt{2}+1} \frac{t}{\left(1+t^{2}\right) \sqrt{1-t^{2}}} d t$

The area of the region between the curves $y=\sqrt{\frac{1+\sin x}{\cos x}}$ and $y=\sqrt{\frac{1-\sin x}{\cos x}}$ and bounded by the
lines $x=0$ and $x=\frac{\pi}{4}$ is $\quad$ [Single Correct Option 2008]
(a) $\int_{0}^{\sqrt{2}-1} \frac{t}{\left(1+t^{2}\right) \sqrt{1-t^{2}}} d t$
(b) $\int_{0}^{\sqrt{2}-1} \frac{4 t}{\left(1+t^{2}\right) \sqrt{1-t^{2}}} d t$
(c) $\int_{0}^{\sqrt{2}+1} \frac{4 t}{\left(1+t^{2}\right) \sqrt{1-t^{2}}} d t$
(d) $\int_{0}^{\sqrt{2}+1} \frac{t}{\left(1+t^{2}\right) \sqrt{1-t^{2}}} d t$

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Key Concepts

Variable Substitution

Variable substitution is an essential technique in integration where a new variable is introduced to simplify the integrand. By substituting a function and its differential, the integrals can often be recast into forms that are more directly integrable, which also requires careful adjustment of the integration limits.

Area Between Curves

This concept involves determining the region enclosed by two graphs. In calculus, the area between curves is found by integrating the difference of the functions over a given interval, which necessitates identifying the correct order of subtraction (upper function minus lower function) across the interval.

Definite Integration

Definite integration is a fundamental tool in calculus used to compute the accumulation of quantities, such as areas, over an interval with specified limits. It requires evaluating an integral between two bounds and often involves transforming or simplifying the integrand for easier computation.

Trigonometric Transformation

Trigonometric transformation involves using identities and algebraic manipulations to rewrite trigonometric expressions into simpler or more convenient forms. This is especially useful in integration when dealing with functions that include square roots, sine, and cosine terms, allowing for the application of standard integration methods.

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