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

The integral of (cos(x)-1)/x is \Ci(x)-ln|x|+C

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

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

Ci (x) - ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

= Ci (x) - ln ∣ x ∣

Add a constant to the solution

= Ci (x) - ln ∣ x ∣ + C

(8.6) (3.6)

\int \frac{csc ( ln ( v ^{2} ) )}{v} d v

1 x

f (x) = 4 sin (x)

6 x y^{^{'}} + 5 y = 20

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