Get the solution to "Evaluate the integrals. The integrals are listed in..."

pancha3

Answered question

2021-10-19

Evaluate the integrals. The integrals are listed in random order so you need to decide which integration technique to use.

\int \tan 3 t d t

insonsipthinye

Skilled2021-10-20Added 83 answers

The given integral is

\int \tan 3 t d t

Note that

\int \tan 3 t d t = \int \frac{\sin 3 t}{\cos 3 t} d t

.
Let

u = \cos 3 t

. This implies that

d u = - 3 \sin 3 t d t

\sin 3 t d t = \frac{- d u}{3}

Use these values in

\tan 3 t d t = \int \frac{\sin 3 t}{\cos 3 t} d t

and evaluate as follows.

\int \tan 3 t d t = \int \frac{\sin 3 t}{\cos 3 t} d t

= \int \frac{- d u}{3 u}

= - \frac{1}{3} \int \frac{d u}{u}

= - \frac{1}{3} \ln (u) + C

= - \frac{1}{3} \ln (\cos 3 t) + C

Thus,

\int \tan 3 t d t = - \frac{1}{3} \ln (\cos 3 t) + C

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