Evaluate the indefinite integral of f(x) = (sin x + cos x)/(sin x - cos x).

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll determine the indefinite integral by changing the variable.

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

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

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william1941 | College Teacher | (Level 3) Valedictorian

Posted on

We have to find the indefinite integral of f(x) = (sin x + cos x)/(sin x - cos x).

First let us express sin x - cos x as t.

So t = sin x - cos x

f(x) = (sin x + cos x) / t

Now dt / dx = cos x + sin x

Or dt = (cos x + sin x) dx

Now f(x) dx = dt / t

Int [ f(x) dx ] = Int [ dt / t ] = ln | t | + C

=> ln | sin x - cos x |

So the required integral is ln|sin x - cos x| + C

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