Verified answer

sinx*tgx= sin^2(x)/cos x

sin^2(x)=1-cos^2(x)

=> sinx*tgx = sin^2(x)/cos x = 1/cosx - cos^2(x)/cos(x) = 1/cosx - cosx

Integral of cosx is sinx.

Integral of 1/cosx is ln(1/cosx - tgx).

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result: Integral(sinx*tgx)= ln(1/cosx - tgx) - sinx