Match the integrals in Column I with the values in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS. | Column I | Column II | | --- | --- | | (A) $\int_{-1}^{1} \frac{d x}{1+x^{2}}$ | (p) $\frac{1}{2} \log \left(\frac{2}{3}\right)$ | | (B) $\int_{0}^{1} \frac{d x}{\sqrt{1-x^{2}}}$ | (q) $2 \log \left(\frac{2}{3}\right)$ | | (C) $\int_{2}^{3} \frac{d x}{1-x^{2}}$ | (r) $\frac{\pi}{3}$ | | (D) $\int_{1}^{2} \frac{d x}{x \sqrt{x^{2}-1}}$ | (s) $\frac{\pi}{2}$ |

**(A)** → **(s)**. **(B)** → **(s)**. **(C)** → **(p)**. **(D)** → **(r)**.

JEE Advanced · 2007 · 08 Apr 2007 · Paper 1 · Definite Integrals

Match the integrals in Column I with the values in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

Column I	Column II
(A) $\int_{- 1}^{1} \frac{d x}{1 + x ^{2}}$	(p) $\frac{1}{2} lo g (\frac{2}{3})$
(B) $\int_{0}^{1} \frac{d x}{1 - x ^{2}}$	(q) $2 lo g (\frac{2}{3})$
(C) $\int_{2}^{3} \frac{d x}{1 - x ^{2}}$	(r) $\frac{π}{3}$
(D) $\int_{1}^{2} \frac{d x}{x x ^{2} - 1}$	(s) $\frac{π}{2}$

Answer: A → IV, B → IV, C → I, D → III

Solution

(A)

\int_{- 1}^{1} \frac{d x}{1 + x ^{2}} = [tan^{- 1} x]_{- 1}^{1} = \frac{π}{4} - (- \frac{π}{4}) = \frac{π}{2}

→ (s).

(B)

\int_{0}^{1} \frac{d x}{1 - x ^{2}} = [sin^{- 1} x]_{0}^{1} = \frac{π}{2}

→ (s).

(C)

\int_{2}^{3} \frac{d x}{1 - x ^{2}} = \frac{1}{2} ln \frac{1 + x}{1 - x}_{2}^{3} = \frac{1}{2} [ln (- 2) - ln (- 1/3)] = \frac{1}{2} ln (2/3)

→ (p).

(D)

\int_{1}^{2} \frac{d x}{x x ^{2} - 1} = [sec^{- 1} x]_{1}^{2} = \frac{π}{3} - 0 = \frac{π}{3}

→ (r).

JEE Advanced · 2007 · 08 Apr 2007 · Paper 1 · Match the columns

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JEE Advanced · 2007 · 08 Apr 2007 · Paper 1 · Definite Integrals

Solution

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