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$\int \frac{x^{2} (x \sec^{2} x + \tan x)}{(x \tan x + 1)^{2}} d x is equal to$

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a

−x2xtanx+1+2logxsinx+cotx+C

b

x2(xtanx+1)2−2logxsinx+cotx+C

c

x2xtanx+1+2logxsinx−cosx+C

d

x2xtanx+1+2logxsinx+cosx+K

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detailed solution

Correct option is A

Let I=∫x2xsec2⁡x+tan⁡x(xtan⁡x+1)2dxUsing integrating by partsI=x2∫xsec2x+tanxxtanx+12dx−∫ddx(x2)∫xsec2x+tanxxtanx+12dxxtan⁡x+1=txsec2⁡x+tan⁡xdx=dtI=x2∫dtt2−∫2x∫dtt2dx=−x2t−∫2x−1tdx=−x2xtan⁡x+1+∫2x(xtan⁡x+1)dx=−x2xtan⁡x+1+∫2xcos⁡xxsin⁡x+cos⁡xdx Put xsin⁡x+cos⁡x=y(xcos⁡x+1sin⁡x−sin⁡x)=−x2xtan⁡x+1+2∫dtt=−x2xtan⁡x+1+2log⁡|xsin⁡x+cos⁡x|+C

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