What is the integral of (tan(t)+sin(t))cos(t) ?

The integral of (tan(t)+sin(t))cos(t) is -cos(t)+(sin^2(t))/2+C

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integral of (tan(t)+sin(t))cos(t)

\int (tan (t) + sin (t)) cos (t) d t

- cos (t) + \frac{sin ^{2} ( t )}{2} + C

Solution steps

\int (tan (t) + sin (t)) cos (t) d t

Rewrite using trig identities

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

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

\int sin (t) cos (t) d t = \frac{sin ^{2} ( t )}{2}

= - cos (t) + \frac{sin ^{2} ( t )}{2}

Add a constant to the solution

= - cos (t) + \frac{sin ^{2} ( t )}{2} + C

What is the integral of (tan(t)+sin(t))cos(t) ?
The integral of (tan(t)+sin(t))cos(t) is -cos(t)+(sin^2(t))/2+C