Question

Evaluate the integral\(\int {\frac{{{\rm{dt}}}}{{{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{t + cos2t}}}}} \).

Short answer

The integral value of the given equation is\(\int {\frac{{{\rm{dt}}}}{{{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{t + cos2t}}}}} {\rm{ = tant + C}}\).

Step-by-step solution

Expand the equation.

For the cosine of a double angle, use identity.

\({\rm{cos(2t) = co}}{{\rm{s}}^{\rm{2}}}{\rm{t - si}}{{\rm{n}}^{\rm{2}}}{\rm{t}}\)

So,

\(\begin{array}{c}\int {\frac{{{\rm{dt}}}}{{{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{t + cos(2t)}}}}} {\rm{ = }}\int {\frac{{{\rm{dt}}}}{{{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{t + co}}{{\rm{s}}^{\rm{2}}}{\rm{t - si}}{{\rm{n}}^{\rm{2}}}{\rm{t}}}}} \\{\rm{ = }}\int {\frac{{{\rm{dt}}}}{{{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{t}}}}} \\{\rm{ = }}\int {{\rm{se}}{{\rm{c}}^{\rm{2}}}} {\rm{tdt}}\\{\rm{ = tant + C}}\end{array}\)

Final answer.

where the formula for the indefinite integral\({\rm{se}}{{\rm{c}}^{\rm{2}}}{\rm{t}}\)of was used in the last equality which is elementary integral.

Therefore, the integral value of the given equation is\(\int {\frac{{{\rm{dt}}}}{{{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{t + cos2t}}}}} {\rm{ = tant + C}}\).