Correct Answer - Option 1 :
\(-\dfrac{1}{4}\)
Concept:
cot2 x = cosec2 x - 1
\(\rm \int x^{n}dx = \frac{x^{n+1}}{n+1} + C\)
Calculation:
I = \(\rm \int_0^{π/4}(\cot^3 x + \cot x)dx \ \)
I = \(\rm\int_{0}^{π /4}\left [ \left ( cosec^{2}x-1 \right )cotx+ cotx \right ]dx\)
I = \(\rm \int_{0}^{π /4}\left [ cosec^{2}x. cotx-cotx+cotx \right ]dx\)
I = \(\rm \int_{0}^{π /4}\left ( cosec^{2}x.cotx \right )dx\)
I = \(\rm \int_{0}^{π /4}\left ( \frac{1}{sin^{2}x} . \frac{cosx}{sinx}\right )dx\)
I = \(\rm\int_{0}^{π/4}\left ( \frac{cosx}{sin^{3}x} \right )dx\) ...(1)
Let , sin x = t
Differentiate both side w.r.t x ,
cos x dx = dt
If , x = 0 , then t = 0
x = π /4 , then t = \(\frac{1}{\sqrt{2}}\)
substitute above values in eq. (1)
I = \(\rm\int_{0}^{\frac{1}{\sqrt{2}}}\left ( \frac{dt}{t^{3}} \right )\)
I = \(\rm \left [ \frac{1}{-2t^{2}} \right ]_{0}^{\frac{1}{\sqrt{2}}}\)
I = \(\rm \frac{-1}{2}\left [ \frac{1}{2}-0 \right ]\)
I = \(\frac{-1}{4}\)
The correct option is 1.