1. \(\int \frac{1 + cos^2x + tan^2x}{sin^2x}dx\)

\(=\int (cosec^2x + cot^2x+sec^2x)dx\)

\(=\int(2cosec^2x-1+sec^2x)dx\,\,\,\,\,\,\,\,\,(\because cot^2x = cosec^2x-1)\)

\(=2\int cosec^2x\,dx-\int dx + \int sec^2x dx\)

= -2 cot x - x + tan x +C

2. \(\int \frac{x^4-8x}{x -2 }dx\)

= \(\int \frac{(x-2)(x^3+2x^2+4x)}{x-2}dx\)

= \(\int (x^3+2x^2+4x)dx\)

= \(\frac{x^4}{4}+\frac{2x^3}{3}+\frac{4x^2}{2}+C\)

= \(\frac{x^4}{4}+\frac23x^3+2x^2 + C\)

3. Let I = \(\int(x+1)\sqrt{2-x}\,\,\,\,\,dx\)

Let 2 - x = f² 

⇒ x = 2 - f²

⇒ -dx = 2tdt

⇒ dx = -2tdt

\(\therefore\) I = \(-\int(2-t^2+1)t.2tdt\)

= \(-2\int(3t^2-t^4)dt\)

= \(-2[3\frac{t^3}3-\frac{t^5}5]+C\)

= \(-2(t^3- \frac{t^5}5)+C\)

= \(-2((2-x)^{\frac32}-\frac15(2-x)^{\frac52})+C\)

= \(-2(2-x)^{\frac32}-\frac25(2-x)^{\frac52}+C\)

\(\therefore\) \(\int(x+1)\sqrt{2-x}\,\,\,\,dx\)  = \(-2(2-x)^{\frac32}-\frac25(2-x)^{\frac52}+C\)

4. Let I = \(\int\frac{cos\theta}{\sqrt{1-sin\theta}}\,\,\,d\theta\)

Put 1 - sinθ = t²

⇒ - cosθ dθ = 2tdt

⇒ cosθ dθ = -2tdt

\(\therefore\) i = \(\int\frac{-2tdt}t= -2\int dt\)

= -2t + C

= \(-2\sqrt{1-sin\theta}+C\)

\(\therefore\) \(\int\frac{cos\theta}{\sqrt{1-sin\theta}}\,\,\,d\theta\) = \(-2\sqrt{1-sin\theta}+C\)