Consider the integral involving tangent and secant

MATHEMATICS · Subjective · A Level · Mathematics · Integration · Integration

Problem

Consider the integral involving tangent and secant. (i) Evaluate the indefinite integral

\displaystyle \int \tan(ax+b)\,\sec^{2}(ax+b)\,dx

for

a\ne 0

. (ii) Hence, compute

K=\displaystyle \int_{0}^{\pi/4}\tan\bigl(2x+\tfrac{\pi}{6}\bigr)\,\sec^{2}\!\bigl(2x+\tfrac{\pi}{6}\bigr)\,dx

. (iii) Express your answer for

K

as a rational multiple of

\tan^{2}\!(\tfrac{\pi}{3})