If ( 1 - 5 cos 2 x sin 5 x ? cos 2 x ) ? d x = f ( x ) + C , where C is the constant of integration, then f ( 6 ) - f ( 4 ) is equal to [2026]

Question

If $\int ((\frac{1 - 5 \cos^{2} x}{\sin^{5} x \cos^{2} x})) d x = f (x) + C,$ where C is the constant of integration, then $f (\frac{π}{6}) - f (\frac{π}{4})$ is equal to [2026]

Smart Achievers · Accepted Answer

(1)

$\int \frac{d x}{\sin^{5} x \cos^{2} x} - 5 \int \frac{d x}{\sin^{5} x}$

$= \int \frac{\sec^{2} x d x}{\sin^{5} x} - 5 \int \frac{d x}{\sin^{5} x}$

$By IBP$

$= \frac{\tan x}{\sin^{5} x} - \int \frac{5}{\sin^{6} x} \cos x \tan x d x - 5 \int \frac{d x}{\sin^{5} x}$

$= \frac{\tan x}{\sin^{5} x} + C$

$f (x) = \frac{\tan x}{\sin^{5} x}$

$f (\frac{π}{6}) - f (\frac{π}{4}) = \frac{2^{5}}{\sqrt{3}} - {(\sqrt{2})}^{5} = 4 \sqrt{2} - \frac{32}{\sqrt{3}}$

$= \frac{32}{\sqrt{3}} - 4 \sqrt{2}$

$= \frac{4}{\sqrt{3}} (8 - \sqrt{6})$

Solution

Q.
If $\int ((\frac{1 - 5 \cos^{2} x}{\sin^{5} x \cos^{2} x})) d x = f (x) + C,$ where C is the constant of integration, then $f (\frac{π}{6}) - f (\frac{π}{4})$ is equal to [2026]

1 $\frac{4}{\sqrt{3}} (8 - \sqrt{6})$

2 $\frac{2}{\sqrt{3}} (4 + \sqrt{6})$

3 $\frac{1}{\sqrt{3}} (26 + \sqrt{3})$

4 $\frac{1}{\sqrt{3}} (26 - \sqrt{3})$

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Solution

Q. If ∫(1-5cos2xsin5x cos2x)dx=f(x)+C, where C is the constant of integration, then f(π6)-f(π4) is equal to [2026]

1 43(8-6)

2 23(4+6)

3 13(26+3)

4 13(26-3)