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

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

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integral of (1-sin(t))/(t+cos(t))

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

ln ∣ t + cos (t) ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = t + cos (t)

= ln ∣ t + cos (t) ∣

Add a constant to the solution

= ln ∣ t + cos (t) ∣ + C

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