Verified answer

I think you mean the substitution t = 7sinx

then dt = 7cosx dx, and dx = dt / 7cosx

you can plug in for x and dx

∫[49-(7sinx)^2]^(1/2) dt/7cosx

simplify the expression inside the brackets

[49 - 49sin^2(x)]^(1/2) = [49(1-sin^2(x)]^(1/2) = 7sqrt(1-sin^2(x))

use the identity cos^2(x)+sin^2(x) = 1, so sqrt(1-sin^2(x)) = cos(x)

term in brackets = 7cosx

∫7cosx * dt/7cosx = ∫dt = t + C = 7sinx + C