What is primitive of function (cosx)^3/sin x?

1 Answer | Add Yours

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to use the evaluate the primitive of the given function, hence, you need to evaluate the indefinite integral of the given function, such that:

`int (cosx)^3/sin x dx = int (cos^2 x)/sin x *cos x dx`

You need to use the Pyrhagorean trigonometric identity, such that:

`cos^2 x = 1 - sin^2 x`

`int (cos^2 x)/sin x *cos x dx = int (1 - sin^2 x)/sin x *cos x dx`

You may solve the indefinite integral using the substitution process, such that:

`sin x = y => cos x dx = dy`

Replacing the variable yields:

`int (1 - sin^2 x)/sin x *cos x dx = int (1 - y^2)/y *dy`

You need to split the integral using the property of linearity of integral, such that:

`int (1 - y^2)/y *dy = int 1/y dy - int y dy`

`int (1 - y^2)/y *dy = ln|y| - y^2/2 + c`

Replacing back `sin x` for `y` yields:

`int (cosx)^3/sin x dx = ln|sin x| - (sin^2 x)/2 + c`

Hence, evaluating the requested primitive yields, under the given conditions, `int (cosx)^3/sin x dx = ln|sin x| - (sin^2 x)/2 + c.`


We’ve answered 318,916 questions. We can answer yours, too.

Ask a question