Evaluate the indefinite integral

Question

∫cos^4(x)dx?

Answer 1

ʃcos⁴xdx =ʃcos²x ×cos²xdx

Recall : Trigonometry identities cos²x = (1 + cos2x ) / 2

                   =ʃ(1 + cos2x) / 2 (1 + cos2x) / 2 dx

                   =1 / 4ʃ(1 + cos2x)² dx

                   =1 / 4ʃ(1 + 2cos2x + cos²(2x))dx

                   =1 / 4ʃ(1 + 2cos2x + (1 + cos4x) / 2) dx

                   =1 / 8ʃ(2 + 4cos2x + 1 + cos4x) dx

                    = 1 / 8ʃ(3 +4cos2x + cos4x) dx

                    = 1/ 8[3x + 4(sin2x) / 2 + 4(sin4x) /4] + C

                    = 1 / 8[3x + 2sin2x + sin4x] + C.

Answer 2

Calculate the integral:

                   

                  

                 

                 

                

                

Answer 3

Using the formulas: