Question Evaluate the integral. (sect tan ti+t cos 2t j +sin? 2t cos 2tk)dt 0

Split the given into three Integral along i, j, k components. Use standard Integral formulae formula for first one, Integration by parts for second one and substitution method for last Integral, Sum up all three for given Integral. {:[int_(0)^(pi//4)(sec t tan t( hat(i))+t cos 2t( hat(j))+sin^(2)2t cos 2t( hat(k)))dt],[={int_(0)^(pi//4)sec t tan tdt} hat(i)+{int_(0)^(pi//4)t cos tdt} hat(j)+],[{int_(0)^(pi//4)sin^(2)2t cos 2t+( hat(}))( hat(k)):}]:}First Integral:{:[int_(0)^(pi//4)sec t tan tdt={sec^(')t}_(0)^(pi//4)],[=sec (pi)/(4)-sec 0=sqrt2-1]:}Second Integral:{:[=(pi)/(8)*sin (pi)/(2)-0-{-(cos 2t)/(4)}_(0)^(pi//4)],[=(pi)/(8)*1-{0+(1)/(4)}=(pi)/(8)-(1)/(4)],[=(pi-2)/(8)]:}Third Integral:{:[" Third Integral: "],[int_(0)^(pi//4)sin^(2)2t cos 2tdt=(1)/(2)*int_(0)^(pi//4)sin^(2)2td(sin 2t)],[=(1)/(2){(sin^(3)2t)/(3)}_(0)^(pi//4)=(1)/(6)[1-0]=(1)/(6)],[:." Solution "=(sqrt2-1) hat(i)+((pi-2)/(8)) hat(j)+(1)/(6) hat(k)]:} ...