(C) : Given,\(\int \cos ^{-3 / 7} \mathrm{x} \sin ^{-11 / 7} \mathrm{x} d \mathrm{x}\) \(=\int \frac{\sin ^{-11 / 7} x}{\cos ^{-11 / 7} \cos ^{2} x} d x=\int \tan ^{-11 / 7} x \sec ^{2} x d x\) Let, \(\mathrm{t}=\tan x\) \(\mathrm{dt}=\sec ^{2} \mathrm{xdx}\) \(=\int(t)^{-11 / 7} \mathrm{dt}\) \(=\frac{(\mathrm{t})^{-11 / 7+1}}{\frac{-11}{7}+1}+\mathrm{C}=\frac{-7}{4}\left[(\mathrm{t})^{-4 / 7}\right]+\mathrm{C}\) \(=-\frac{7}{4} \tan ^{-4 / 7} \mathrm{x}+\mathrm{C}\)
Manipal-2016
Integral Calculus
86333
If \(\int \sin \left\{2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right\} d x\) \(=A \sin ^{-1} x+B x \sqrt{1-\mathbf{x}^{2}}+C\), then \(A+B\) is equal to
(C) : Given,\(\int \cos ^{-3 / 7} \mathrm{x} \sin ^{-11 / 7} \mathrm{x} d \mathrm{x}\) \(=\int \frac{\sin ^{-11 / 7} x}{\cos ^{-11 / 7} \cos ^{2} x} d x=\int \tan ^{-11 / 7} x \sec ^{2} x d x\) Let, \(\mathrm{t}=\tan x\) \(\mathrm{dt}=\sec ^{2} \mathrm{xdx}\) \(=\int(t)^{-11 / 7} \mathrm{dt}\) \(=\frac{(\mathrm{t})^{-11 / 7+1}}{\frac{-11}{7}+1}+\mathrm{C}=\frac{-7}{4}\left[(\mathrm{t})^{-4 / 7}\right]+\mathrm{C}\) \(=-\frac{7}{4} \tan ^{-4 / 7} \mathrm{x}+\mathrm{C}\)
Manipal-2016
Integral Calculus
86333
If \(\int \sin \left\{2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right\} d x\) \(=A \sin ^{-1} x+B x \sqrt{1-\mathbf{x}^{2}}+C\), then \(A+B\) is equal to
(C) : Given,\(\int \cos ^{-3 / 7} \mathrm{x} \sin ^{-11 / 7} \mathrm{x} d \mathrm{x}\) \(=\int \frac{\sin ^{-11 / 7} x}{\cos ^{-11 / 7} \cos ^{2} x} d x=\int \tan ^{-11 / 7} x \sec ^{2} x d x\) Let, \(\mathrm{t}=\tan x\) \(\mathrm{dt}=\sec ^{2} \mathrm{xdx}\) \(=\int(t)^{-11 / 7} \mathrm{dt}\) \(=\frac{(\mathrm{t})^{-11 / 7+1}}{\frac{-11}{7}+1}+\mathrm{C}=\frac{-7}{4}\left[(\mathrm{t})^{-4 / 7}\right]+\mathrm{C}\) \(=-\frac{7}{4} \tan ^{-4 / 7} \mathrm{x}+\mathrm{C}\)
Manipal-2016
Integral Calculus
86333
If \(\int \sin \left\{2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right\} d x\) \(=A \sin ^{-1} x+B x \sqrt{1-\mathbf{x}^{2}}+C\), then \(A+B\) is equal to
(C) : Given,\(\int \cos ^{-3 / 7} \mathrm{x} \sin ^{-11 / 7} \mathrm{x} d \mathrm{x}\) \(=\int \frac{\sin ^{-11 / 7} x}{\cos ^{-11 / 7} \cos ^{2} x} d x=\int \tan ^{-11 / 7} x \sec ^{2} x d x\) Let, \(\mathrm{t}=\tan x\) \(\mathrm{dt}=\sec ^{2} \mathrm{xdx}\) \(=\int(t)^{-11 / 7} \mathrm{dt}\) \(=\frac{(\mathrm{t})^{-11 / 7+1}}{\frac{-11}{7}+1}+\mathrm{C}=\frac{-7}{4}\left[(\mathrm{t})^{-4 / 7}\right]+\mathrm{C}\) \(=-\frac{7}{4} \tan ^{-4 / 7} \mathrm{x}+\mathrm{C}\)
Manipal-2016
Integral Calculus
86333
If \(\int \sin \left\{2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}}\right\} d x\) \(=A \sin ^{-1} x+B x \sqrt{1-\mathbf{x}^{2}}+C\), then \(A+B\) is equal to
