What is the integral of ((1-tcos(t)))/t ?

The integral of ((1-tcos(t)))/t is ln|t|-sin(t)+C

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integral of ((1-tcos(t)))/t

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

ln ∣ t ∣ - sin (t) + C

Solution steps

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

Expand

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

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

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{1}{t} d t - \int cos (t) d t

\int \frac{1}{t} d t = ln ∣ t ∣

\int cos (t) d t = sin (t)

= ln ∣ t ∣ - sin (t)

Add a constant to the solution

= ln ∣ t ∣ - sin (t) + C

What is the integral of ((1-tcos(t)))/t ?
The integral of ((1-tcos(t)))/t is ln|t|-sin(t)+C