A) \[\log |{{\sin }^{4/7}}x|+c\]
B) \[\frac{4}{7}{{\tan }^{4/7}}x+c\]
C) \[\frac{-7}{4}{{\tan }^{-4/7}}x+c\]
D) \[\log |{{\cos }^{3/7}}x|+c\]
E) \[\frac{7}{4}{{\tan }^{-4/7}}x+C\]
Correct Answer: C
Solution :
\[m+n=-\frac{3}{7}+\left( \frac{-11}{7} \right)=-2\] (?ve integer) \[I=\int{{{\cos }^{-3/7}}x\left( {{\sin }^{(-2+3/7)}}x \right)dx}=\int{{{\cos }^{-3/7}}}x\,{{\sin }^{-2}}x\,{{\sin }^{3/7}}xdx\] \[=\int{\frac{\cos e{{c}^{2}}x}{\left( \frac{{{\cos }^{3/7}}x}{{{\sin }^{3/7}}x} \right)}}dx=\int{\frac{\cos e{{c}^{2}}x\,dx}{{{\cot }^{3/7}}x}}\] Put \[\cot x=t\]Þ \[-\cos e{{c}^{2}}xdx=dt\] \[I=-\int{\frac{dt}{{{t}^{3/7}}}}\]\[=-\frac{{{t}^{-\frac{3}{7}+1}}}{-\frac{3}{7}+1}+c\]\[=-\frac{7}{4}{{t}^{4/7}}+c\] \[=-\frac{7}{4}{{\cot }^{4/7}}x+c\]\[=-\frac{7}{4}{{\tan }^{-4/7}}x+c\].
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