Match List-I with List-II | List-I | List-II | |---|---| | Definite integral | Value | | (A) _0^1 2x/1 + x^2 dx | (I) 2 | | (B) _-1^1 sin^3 x ^4 x dx | (II) log_e(3/2) | | (C) _0^ x dx | (III) log_e 2 | | (D) _2^3 2/x^2 - 1 dx | (IV) 0 | Choose the correct answer from the options given below: Option 1: (A) - (I), (B) - (II), (C) - (III), (D) - (IV) Option 2: (A) - (III), (B) - (II), (C) - (I), (D) - (IV) Option 3: (A) - (III), (B) - (I), (C) - (IV), (D) - (II) Option 4: (A) - (III), (B) - (IV), (C) - (I), (D) - (II)

The answer is '(A) - (III), (B) - (IV), (C) - (I), (D) - (II)'. View full solution:

Calculus > Integrals

Medium

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Match List-I with List-II | List-I | List-II | |---|---| | Definite integral | Value | | (A) $\int_0^1 \frac{2x}{1 + x^2} dx$ | (I) 2 | | (B) $\int_{-1}^1 sin^3 x \cos^4 x dx$ | (II) $log_e\left(\frac{3}{2}\right)$ | | (C) $\int_0^{\pi} \sin x dx$ | (III) $log_e 2$ | | (D) $\int_2^3 \frac{2}{x^2 - 1} dx$ | (IV) 0 | Choose the correct answer from the options given below:

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (III), (B) - (II), (C) - (I), (D) - (IV)

(A) - (III), (B) - (I), (C) - (IV), (D) - (II)

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

✅ Correct Option: 4

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$\int \frac{x^2 + 4}{(x^2 + 3)(x^2 + 5)} \, dx = u \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) + v \tan^{-1}\left(\frac{x}{\sqrt{5}}\right) + c$ , where c is arbitrary constant, then value of $\frac{1}{u^2} + \frac{1}{v^2}$ is equal to :

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