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First separate the variables so that the left side of the equation is an expression only in terms of y and the right side only in terms of x.exp(y)dy=sin2x dxSecondly both sides have to be integrated to o...

2^{3x - 1} = 3log₂(2^3x-1) = log₂(3)3x - 1 = log₂(3)3x = 1 + log₂(3)x = ¹/₃ + ¹/₃log₂(...

Apply the double angle formula to cos2x to yield the requested result.
cos2x = 2cos^2(x) - 1
Spot that the question asks us to prove the value of cos^2(x) when integrated, and that we can move t...

First, manoeuvre variables so that we can integrate the equation.
1/(x-6)^(1/2) dx = -2 dt
Integrate the equation and add the constant.
2(x-6)^(1/2) = -2t +c
Solve for t.
t = -(x-...

Using the substitution u = sec(z)=> du = sec(z)tan(z) dz.So, the integral ∫ y dz = ∫ sec(z)tan(z)/sqrt(sec(z)) dz=> ∫ y dz = ∫ 1/sqrt(u) du = 2sqrt(u) + C = 2sqrt(sec(z)) + C.