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The value of the integral $\int_{0}^{3} \frac{d x}{\sqrt{x + 1} + \sqrt{5 x + 1}}$ is

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a

11/15

b

14/15

c

2/5

d

none of these

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detailed solution

Correct option is D

. The integrand can be written as x+1−5x+1−4xSo the given integral is of the form I1+I2 I1=−14∫03 x+1xdx=−12∫12 t2t2−1dt=−12∫12 1+1t2−1dt=−121−12limt→1 log⁡|t−1|t+1x+1=t2and I2=14∫03 5x+1xdx=14∫14 2u2u2−1duu2=5x+1=123+12log⁡35−12limu→1 log⁡|u−1|u+1Thus I1+I2=1+14log⁡35

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