Evaluate ( 1 + cos ( 2x) ) 3/2 dx . Solution. First, use the identity 1 + cos(2x) = 2(cos(x)) to transforme the integral. We obtain: It/2 It/2 (1 + cos(2x))3/2 dx = dx Next, find an antiderivative of the new integrand function: (1 + cos(2x)) 32 dx = +C where C is a constant. Finally, deduce the value of the definite integral: (1 + cos(2x))3/2 dx =tan(2 In(x)) Use substitution to evaluate the integral dx. 4x Solution. Set u = du = dx Then, we obtain tan(2 In(x)) dx : du 4x +C + a function in the variable u +C a function in the variable x where C is a constantsinh(x) Calculate dx 4 cosh(x) + 5 Solution. Set u = du = dx Then, we obtain sinh(x) dx = du 4 cosh(x) + 5 +C a function in the variable = u +C a function in the variable X where C is a constant