\(\text { Let } \sqrt{x}=t\)
\(\text { Then } x=t^{2}\)
\(\therefore \frac{d x}{d t}=2 t\)
\(\therefore d x=2 t d t\)
\(\therefore \int \sin \sqrt{x} d x=\int \sin t \times 2 t d t\)
\(=2 \int t \sin t d t\)
\(\text { Applying Bernoulli's formula }\)
\(=2[t(-\cos t)-1(-\sin t)]+c\)
\(\text { Putting } t=\sqrt{x}\)
\(=2[-\sqrt{x} \cos \sqrt{x}+\sin \sqrt{x}]+c\)

\(\int \frac{x+2}{\sqrt{x^{2}-1}} d x =\int\left(\frac{x}{\sqrt{x^{2}-1}} d x+\frac{2}{\sqrt{x^{2}-1}}\right) d x \)
\(=\sqrt{x^{2}-1}+2 \log \left|x+\sqrt{x^{2}-1}\right|+c \)

WKT
\(\int e^{a x} \cos b x d x=\frac{e^{a x}}{a^{2}+b^{2}}[a \cos b x+b \sin b x]+c\)
\(\text { Putting } a=-4 \text { and } b=1\)
\(\int e^{-4 x} \cos x d x=\frac{e^{-4 x}}{17}[-4 \cos x+\sin x]+c\)

\(\int \frac{x^{2}+\cos ^{2} x}{x^{2}+1} \operatorname{cosec}^{2} x d x \)
\(=\int \frac{x^{2}+\left(1-\sin ^{2} x\right)}{x^{2}+1} \times \frac{1}{\sin ^{2} x} d x \)
\(=\int \frac{\left(x^{2}+1\right)-\sin ^{2} x}{\left(x^{2}+1\right) \sin ^{2} x} d x \)
\(=\int\left(\frac{1}{\sin ^{2} x}-\frac{1}{x^{2}+1}\right) d x \)
\(=\int\left(\operatorname{cosec}^{2} x-\frac{1}{x^{2}+1}\right) d x \)
\(=-\cot x-\tan ^{-1} x+c \)

\(\sin ^{8} x-\cos ^{8} x =\left(\sin ^{4} x-\cos ^{4} x\right)\left(\sin ^{4} x+\cos ^{4} x\right) \)
\(=\left(\sin ^{2} x+\cos ^{2} x\right)\left(\sin ^{2} x-\cos ^{2} x\right) \)
\(\times \left[\left(\sin ^{2} x+\cos ^{2} x\right)^{2}-2 \sin ^{2} x \cos ^{2} x\right] \)
\(=(1)(-\cos 2 x)\left(1-2 \sin ^{2} x \cos ^{2} x\right) \)
\( \therefore \frac{\sin ^{8} x-\cos ^{8} x}{1-2 \sin ^{2} x \cos ^{2} x}=-\cos 2 x \)
\( \therefore \int \frac{\sin ^{8} x-\cos ^{8} x}{1-2 \sin ^{2} x \cos ^{2} x} d x=\frac{-\sin 2 x}{2}+c \)

\(\int \frac{e^{6 \log x}-e^{5 \log x}}{e^{4 \log x}-e^{3 \log x}} d x =\int \frac{x^{6}-x^{5}}{x^{4}-x^{3}} d x \)
\(=\int \frac{x^{2}\left(x^{4}-x^{3}\right)}{x^{4}-x^{3}} d x \)
\(=\int x^{2} d x=\frac{x^{3}}{3}+c \)