Match List-I with List-II
|
List-I Integral |
List-II Solution: C is an arbitrary constant |
|
(A) $\int\frac{dx}{x^2+25}$ |
(I) $\frac{1}{10}\log\left|\frac{5+x}{5-x}\right|+C$ |
|
(B) $\int\frac{dx}{x^2-25}$ |
(II) $\log|x+\sqrt{x^2-25}|+C$ |
|
(C) $\int\frac{dx}{25-x^2}$ |
(III) $\frac{1}{5}\tan^{-1}(\frac{x}{5})+C$ |
|
(D) $\int \frac{dx}{\sqrt{x^{2}-25}}$ |
(IV) $\frac{1}{10}\log\left|\frac{5-x}{5+x}\right|+C$ |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
|
List-I Integral |
List-II Solution: C is an arbitrary constant |
|
(A) $\int\frac{dx}{x^2+25}$ |
(III) $\frac{1}{5}\tan^{-1}(\frac{x}{5})+C$ |
|
(B) $\int\frac{dx}{x^2-25}$ |
(IV) $\frac{1}{10}\log\left|\frac{5-x}{5+x}\right|+C$ |
|
(C) $\int\frac{dx}{25-x^2}$ |
(I) $\frac{1}{10}\log\left|\frac{5+x}{5-x}\right|+C$ |
|
(D) $\int \frac{dx}{\sqrt{x^{2}-25}}$ |
(II) $\log|x+\sqrt{x^2-25}|+C$ |
(A) $\int \frac{dx}{x^{2}+25}$
Standard form: $\int \frac{dx}{x^{2}+a^{2}} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+C$
Here $a=5$ ⇒ $\frac{1}{5}\tan^{-1}\left(\frac{x}{5}\right)+C$ → (III)
(B) $\int \frac{dx}{x^{2}-25}$
Standard form: $\int \frac{dx}{x^{2}-a^{2}} = \frac{1}{2a}\log\left|\frac{x-a}{x+a}\right|+C$
Here $a=5$ ⇒ $\frac{1}{10}\log\left|\frac{5-x}{5+x}\right|+C$ → (IV)
(C) $\int \frac{dx}{25-x^{2}}$
Standard form: $\int \frac{dx}{a^{2}-x^{2}} = \frac{1}{2a}\log\left|\frac{a+x}{a-x}\right|+C$
Here $a=5$ ⇒ $\frac{1}{10}\log\left|\frac{5+x}{5-x}\right|+C$ → (I)
(D) $\int \frac{dx}{\sqrt{x^{2}-25}}$
Standard form: $\int \frac{dx}{\sqrt{x^{2}-a^{2}}} = \log|x+\sqrt{x^{2}-a^{2}}|+C$
Here $a=5$ ⇒ $\log|x+\sqrt{x^{2}-25}|+C$ → (II)