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How to solve Cos^5 or sin, or similiar (integrals)

hey guys, I have uploaded the problem below

The problem is $cos^5x dx

The main part I don't understand is how is the symmetry is done for the exponentials.

right the constants are taken from the pascal triangle. Why is there a part in the formula multiplied with a postive e function, multiplied with a negative

Please **don't** just tell me to study they book, because I tried that before coming here, and there are no examples on this in ours, they have just given the formula, and thats it, they expect us to know this from highschool. The strange thing here in sweden is that you basically can be admitted to an engineering program without studying complex numbers (math e) I just studied Math D which includes derivatives and basic integrals. They are going to change the admission requirements for engineering programs from autumn this year, so that students with Math E only can get admitted, due to the lack of knowledge... Sucks i know, but what else can I do than coming here.

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