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Sign of Trigonometric Ratios in Different Quadrants

Evaluate : ...

Question

Evaluate : $\int \sqrt{\frac{a - x}{x}} d x$

A

\sqrt{a - x} \sqrt{x} + a {tan}^{- 1} (\frac{\sqrt{x}}{\sqrt{a - x}}) + c

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B

\frac{\sqrt{a - x}}{\sqrt{x}} + a {tan}^{- 1} (\frac{\sqrt{x}}{\sqrt{a - x}}) + c

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C

\frac{\sqrt{a - x}}{\sqrt{x}} - a {tan}^{- 1} (\frac{\sqrt{x}}{\sqrt{a - x}}) + c

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D

\sqrt{a - x} \sqrt{x} - a {tan}^{- 1} (\frac{\sqrt{x}}{\sqrt{a - x}}) + c

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Solution

The correct option is A $\sqrt{a - x} \sqrt{x} + a {tan}^{- 1} (\frac{\sqrt{x}}{\sqrt{a - x}}) + c$
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Similar questions

\int \frac{(x^{2} - 1)}{(x + 1)^{2} \sqrt{x (x^{2} + x + 1)}} d x = A {tan}^{- 1} (\sqrt{\frac{x^{2} + x + 1}{x}}) + c

c

is a constant, then

A =

sin [a {tan}^{- 1} {\sqrt{\frac{b - x}{1 + x}}}] = \sqrt{1 - x^{c}}

.
Find the value of a,b and c.

\int \frac{d x}{\sqrt{(x - α) (β - x)}} = A {tan}^{- 1} f (x) + C

\int \frac{(x - 1)^{2}}{x^{4} + x^{2} + 1} d x = \frac{1}{\sqrt{a}} {tan}^{- 1} (\frac{x^{2} - 1}{x \sqrt{3}}) - \frac{b}{\sqrt{a}} {tan}^{- 1} (\frac{2 x^{2} + 1}{\sqrt{3}}) + C,

then which of the following is/are CORRECT?
(where

a, b

are fixed constants and

C

is integration constant)

\int \frac{2 x^{2} + a^{2}}{x^{2} (x^{2} + a^{2})} d x = \frac{k}{x} + \frac{1}{a} t a n^{- 1} \frac{x}{a} + c

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