A) \[\frac{\cos x}{{{(\sin x)}^{7}}+c}\]
B) \[\frac{\tan x}{{{(\sin x)}^{7}}+c}\]
C) \[\frac{\sec x}{{{(cosx)}^{7}}+c}\]
D) None of these
Correct Answer: B
Solution :
[b] \[I=\int{\frac{{{\sec }^{2}}x-7}{{{\sin }^{7}}x}}dx\] \[=\int{\frac{{{\sec }^{2}}x}{{{\sin }^{7}}x}}dx-7\int{\frac{1}{{{\sin }^{7}}x}}dx\] \[=\frac{\tan x}{{{\sin }^{7}}x}-\int{(-7){{\sin }^{-8}}x.\cos x.tan\,x\,\,dx-7\int{\frac{1}{{{\sin }^{7}}x}dx+C}}\] [Integration by parts \[=\frac{\tan x}{{{\sin }^{7}}x}+7\int{\frac{1}{{{\sin }^{7}}x}dx-7\int{\frac{1}{{{\sin }^{7}}x}}dx+C}\] \[=\frac{\tan x}{{{\sin }^{7}}x}+C\]
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