`int (1+cos(alpha))/sin(alpha) dalpha` Find the indefinite integral

Given to solve,

`int (1+cos(alpha))/sin(alpha) d alpha`

let `alpha = x` (just for convenience)

so,

`int (1+cos(alpha))/sin(alpha) d alpha`

= `int (1+cos(x))/sin(x) dx`

= `int (1/sinx)dx +int cos(x)/sin(x) dx`

=` int cscx dx+ int cot(x) dx`

we know from the general formulas that

`int cscx dx = ln(tan(x/2))`

and

`int cot(x) dx = ln(sinx)`

so ,

`int cscx dx+ int cot(x) dx = ln(tan(x/2)) + ln(sinx) +c`

but `x= alpha `

so,

`int (1+cos(alpha))/sin(alpha) d alpha`

`=ln(tan((alpha)/2)) + ln(sin(alpha)) +c`

Comments

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