Chapter 4

$$ y=e^{\sqrt{\frac{1-x}{1+x}}} $$

$$ y=\sqrt{a^{2}-x^{2}}-a \cos ^{-1} \frac{x}{a} $$

$$ y=\sqrt{x^{2}+1}-\ln \left(\frac{1}{x}+\sqrt{1+\frac{1}{x^{2}}}\right) $$

$$ y=\frac{\sin ^{2} x}{1+\cot x}+\frac{\cos ^{2} x}{1+\tan x} $$

$$ y=\ln \left(x+\sqrt{x^{2}-1}\right)-\frac{x}{\sqrt{x^{2}-1}} $$

$$ y=e^{x}(\sin 3 x-3 \cos 3 x) $$

$$ y=3 x^{3} \sin ^{-1} x+\left(x^{2}+2\right) \sqrt{1-x^{2}} $$

$$ y=\frac{1}{\sqrt{1+e^{-\sqrt{x}}}} $$

$$ y=2 \sin ^{-1} \frac{x-2}{\sqrt{6}}-\sqrt{2+4 x-x^{2}} $$

$$ y=\ln \left(e^{x} \cos x+e^{-x} \sin x\right) $$

$$ y=\frac{1+x \tan ^{-1} x}{\sqrt{1+x^{2}}} $$

$$ y=\frac{1}{\cos (x-\cos x)} $$

$$ y=e^{x} \sin x \cos ^{3} x $$

$$ y=\sqrt[11]{9+6 \sqrt[5]{x^{9}}} $$

$$ y=x-\ln \left(2 e^{x}+1+\sqrt{e^{2 x}+4 e^{x}+1}\right) $$

$$ y=e^{\tan ^{-1} \sqrt{1+\ln (2 x+3)}} $$

$$ y=\frac{e^{x^{2}}}{e^{x}+e^{-x}} $$

$$ y=\ln \tan \frac{x}{2}-\cot x \ln (1+\sin x)-x $$

$$ y=2 \ln \left(2 x-3 \sqrt{1-4 x^{2}}\right)-6 \sin ^{-1} 2 x $$

$$ y=\frac{3 x^{2}-1}{3 x^{3}}+\ln \sqrt{1+x^{2}}+\tan ^{-1} x $$

$$ y=\frac{1}{2}(3-x) \sqrt{1-2 x-x^{2}}+2 \sin ^{-1} \frac{x+1}{\sqrt{2}} $$

$$ y=\ln \left(x \sin x \sqrt{1-x^{2}}\right) $$

$$ y=\frac{\sqrt{x+2}(3-x)^{4}}{(x+1)^{5}} $$

$$ y=\sqrt[5]{\left(1+x e^{\sqrt{x}}\right)^{3}} $$

$$ y=\frac{1}{\sqrt{x}} e^{x^{2}-\tan ^{-1} x+\frac{1}{2} \ln x+1} $$

$$ y=\frac{\sin x}{4 \cos ^{4} x}+\frac{3 \sin x}{8 \cos ^{2} x}+\frac{3}{8} \ln \frac{1+\tan \frac{x}{2}}{1-\tan \frac{x}{2}} $$

$$ y=\frac{x e^{x} \tan ^{-1} x}{\ln ^{5} x} $$

$$ y=\frac{\left(1-x^{2}\right) e^{3 x-1} \cos x}{\left(\cos ^{-1} x\right)^{3}} $$

$$ y=x \sqrt{\left(x^{2}+a^{2}\right)^{3}}+\frac{3 a^{2} x}{2} \sqrt{x^{2}+a^{2}}+\frac{3 a^{4}}{2} \ln \left(x+\sqrt{x^{2}+a^{2}}\right) $$

$$ y=x\left(\sin ^{-1} x\right)^{2}-2 x+2 \sqrt{1-x^{2}} \sin ^{-1} x $$

$$ y=\ln \cos \left(\tan ^{-1} \frac{e^{x}-e^{-x}}{2}\right) $$

$$ y=\frac{1}{m \sqrt{a b}} \tan ^{-1}\left(e^{m x} \sqrt{\frac{a}{b}}\right) $$

$$ y=\frac{1}{3} \ln \frac{x+1}{\sqrt{x^{2}-x+1}}+\frac{1}{\sqrt{3}} \tan ^{-1} \frac{2 x-1}{\sqrt{3}} $$

$$ y=\ln \frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}+2 \tan ^{-1} \sqrt{\frac{1-x}{1+x}} $$

$$ y=\sqrt[3]{\frac{x-5}{\sqrt[5]{x^{2}+4}}} $$

$$ y=\ln \sqrt[4]{\frac{x^{2}+x+1}{x^{2}-x+1}}+\frac{1}{2 \sqrt{3}}\left(\tan ^{-1} \frac{2 x+1}{\sqrt{3}}+\tan ^{-1} \frac{2 x-1}{\sqrt{3}}\right) $$

$$ \left.y=\cos ^{-1} \frac{x^{2 n}-1}{x^{2 n}+1} \text { \\{ Ans. }-\frac{2 n x^{n-1}}{x^{2 n}+1} \text { if } n \text { is even and }-\frac{2 n x^{n}}{|x|\left(x^{2 n}+1\right)}, \text { if } n \text { is odd. }\right\\} $$

$$ y=-\frac{x}{1+8 x^{3}}+\frac{1}{12} \ln \frac{(1+2 x)^{2}}{1-2 x+4 x^{2}}+\frac{\sqrt{3}}{6} \tan ^{-1} \frac{4 x-1}{\sqrt{3}} $$

$$ y=(\sin x)^{\cos x} $$

$$ y=(\tan 2 x)^{\cot ^{\frac{x}{2}}} $$

$$ y=(\ln x)^{x} $$

$$ y=(x+1)^{\frac{2}{2}} $$

$$ y=\frac{(x-2)^{2} \sqrt[3]{x+1}}{(x-5)^{3}} $$

$$ y=x^{\ln x} $$

$$ y=\frac{(x+1) \sqrt[3]{x-2}}{\sqrt[5]{(x-3)^{2}}} $$

$$ y=\sqrt{x \sin x \sqrt{1-e^{x}}} $$

$$ y=\sqrt{\frac{1-\sin ^{-1} x}{1+\sin ^{-1} x}} $$

$$ y=x^{\frac{1}{x}} $$

$$ y=\left(\frac{x}{1+x}\right)^{x} $$

$$ y=2 x^{\sqrt{x}} $$