Evaluate tan$6(8$x$)$ sec$4(8$x$)$ dx$.$SolutionIf we separate one sec$2(8$x$)$ factor, we can express the remaining sec$2(8$x$)$ factor in terms of tangent using the identity sec$2(8$x$)=1$ tan$2(8$x$).$ We can then evaluate the integral by substituting u $=$ tan$(8$x$)$ so that du $=$ dx$.$tan$6(8$x$)$ sec$4(8$x$)$ dx$=$tan$6(8$x$)$ sec$2(8$x$)$ sec$2(8$x$)$ dx$=$tan$6(8$x$)(1$ tan$2(8$x$))$ sec$2(8$x$)$ dx$=18$u$6$ du$=18$u$6$ du$=$ u$972$ C$=172$ tan$9(8$x$)$ C