Evaluate the integral csc2 7 cot 7 d x 7 tanx 72 dx Soluti

Evaluate the integral. csc2 7 cot 7 d x / 7 tan(x / 7)2 dx

Solution

1. Integral of (csc^2(7theta)cot(7theta) dtheta first we will apply the substitution as u = 7theta du = 7 dtheta du/7 = dtheta applying the substitution, we will get Integral of (csc^2(u)cot(u) du/7 now, use the substitution as s = cot(theta) ds = -csc^2(u)du we will get 1/7( Integral of (sds)) Using power rule of integration, we will get 1/7(s^2/2) + c using reverse substitution will give (1/14)(cot^2(7theta)) + c 2.integral of (x/7)(tan(x/7)^2)dx taking the substitution as u = x/7 du = 1/7 dx 7du = dx applying the substitution, we will get integral of 7(u)(tan(u)^2)du 7(integral of (u)(tan(u)^2)du ) 7(-u^2/2 +utan(u) + logcos(u)) + c using the reverse substitution, we will get 7(-x^2/98 + x/7tan(x/7) + logcos(x/7)) + c 3.integral of (21e^sqrt(7x)/2sqrt(x))dx we can rewrite it as 21/2(integral of (e^sqrt(7)sqrtx)/sqrt(x))dx) using the substitution u = sqrt7sqrtx du = sqrt7/2sqrtx dx we will get 3sqrt(7)(integral of e^u du) we know the integral of e^u = e^u so, 3sqrt(7)e^u + c using the reverse substitution 3sqrt(7)e^sqrt7sqrtx + c Hope this will help you!