A) \[\frac{\tan x}{{{\sin }^{7}}x}+C\]
B) \[\frac{\cos x}{{{\sin }^{7}}x}+C\]
C) \[\frac{\sin x}{{{\cos }^{7}}x}+C\]
D) \[\frac{\sin x}{{{\tan }^{7}}x}+C\]
Correct Answer: A
Solution :
Let \[I=\int_{{}}^{{}}{\frac{{{\sec }^{2}}x-7}{{{\sin }^{7}}x}}dx\] \[=\int_{{}}^{{}}{\frac{{{\sin }^{7}}x{{\sec }^{2}}x-7{{\sin }^{7}}x}{{{({{\sin }^{7}}x)}^{2}}}}dx\] \[=\int_{{}}^{{}}{\frac{d}{dx}\left( \frac{\tan x}{{{\sin }^{7}}x} \right)}dx\] \[=\frac{\tan x}{{{\sin }^{7}}x}+C\]
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