Question

integration of sin power 3xoo(​

Answer 1

Explanation:

Integrating the third power of sin() (or any odd power, for that matter), is an easy task (unlike ∫sin2(), which requires a little trick). All you have to do is write the expression as sin()⋅(even power of sin), rewrite the even power using the formula sin2()=1−cos2(), and apply the substitution =cos() (i.e. =−sin()). Let’s see it in practice:

∫sin3()=∫sin()sin2()=∫sin()(1−cos2())=∫−(1−2)|=cos()=−+33+=13cos3()−cos()+.

The same approach works for any odd power of sin() (or cos()), only the resulting expressions get slightly more complicated. For example,

∫sin5()=∫sin()(sin2())2=∫sin()(1−cos2())2=∫−(1−2)2=∫(−4+22−1)=−55+233−+=−15cos5()+23cos3()−cos()+.

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