Solve the integral integrate of (cos5x)/(3+sin5x) dx

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should use substitution to solve the integral such that:

`3 + sin 5x = t => 5cos 5x dx = dt => cos 5x dx = (dt)/5`

Changing the variable yields:

`int (cos 5x)/(3 + sin 5x) dx = int ((dt)/5)/t`

`int ((dt)/5)/t = (1/5)int (dt)/t = (1/5)ln|t| + c`

Substituting back `3 + sin 5x`  for `t`  yields:

`int (cos 5x)/(3 + sin 5x) dx = ln |3 + sin 5x| + c`

Since -`1 =< sin 5x =< 1 => 3 + sin 5x >=0 => |3 + sin 5x| = 3 + sin 5x`

Hence, evaluating the given integral using substitution yields `int (cos 5x)/(3 + sin 5x) dx = ln (3 + sin 5x) + c.`

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