What is the integral of (1/x)(cos(x)-sin(x)) ?

The integral of (1/x)(cos(x)-sin(x)) is \Ci(x)-\Si(x)+C

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integral of (1/x)(cos(x)-sin(x))

integral

(\frac{1}{x}) (cos (x) - sin (x))

Ci (x) - Si (x) + C

Solution steps

\int \frac{1}{x} (cos (x) - sin (x)) d x

Simplify

\frac{1}{x} (cos (x) - sin (x)) : \frac{cos ( x ) - sin ( x )}{x}

= \int \frac{cos ( x ) - sin ( x )}{x} d x

Expand

\frac{cos ( x ) - sin ( x )}{x} : \frac{cos ( x )}{x} - \frac{sin ( x )}{x}

= \int \frac{cos ( x )}{x} - \frac{sin ( x )}{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{cos ( x )}{x} d x - \int \frac{sin ( x )}{x} d x

\int \frac{cos ( x )}{x} d x = Ci (x)

\int \frac{sin ( x )}{x} d x = Si (x)

= Ci (x) - Si (x)

Add a constant to the solution

= Ci (x) - Si (x) + C

x cos (x) cos (2 x)

x^{- 6}

\frac{( x + 7 )}{( x + 9 )}

sin (1 - x)

cos (y^{2})

What is the integral of (1/x)(cos(x)-sin(x)) ?
The integral of (1/x)(cos(x)-sin(x)) is \Ci(x)-\Si(x)+C