What is the indefinite integral of ( sinx + cos x )/ ( sinx -cosx ) ?

neela | High School Teacher | (Level 3) Valedictorian

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To find the integral of (sinx+cosx)/(sinx-cosx).

We put sinx-cosx = t.......(1)

We differentiate sinx-cosx with respect to t.

cosx -(-sinx) = dt

(cosx+sinx)dx = dt.....(2)

Therefore Int {(sinx+cosx)/(sinx - cosx)}dx = Int(1/t) dt = logt +constant.

Int {(sinx +cosx)dt/(sinx-cosx)}dx = log(sinx-cosx) +C, where C is a constant of integration.

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To determine the indefinite integral when the integrand is the given function, we'll use substitution technique.

We'll change the variable x, substituting the denominator by another variable, t.

We'll note the denominator sin x - cos x = t(x)

We'll differentiate the denominator:

(sin x - cos x)' = [cos x - (-sin x)]dx

(cos x + sin x)dx = dt

We'll notice that the numerator of the function is the result of differentiating the function.

We'll calculate the integral:

Int f(x) = Int dt/t

Int dt/t = ln |t| + C

But t = sin x - cos x

The indefinite integral of (sinx + cos x)/ (sinx -cosx) is:

Int f(x) = ln|sin x - cos x| + C