Question

integration of tan3 x​

Answer 1

Answer:

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Step-by-step explanation:

A2A

I=∫tan(3x)dx

I=∫sin(3x)cos3xdx

substitude:

cos(3x)=u

−3×sin(3x)×dx=du

sin(3x)×dx=−du3

apply this conclusion to our integration:

I1=∫−13×1udu

I1=−13×ln|u|+C1,C1∈R

I=−13×ln|cos(3x)|+C,C∈R

Answer 2

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Integral of tan3(x)

Use what you have learned to integrate the function tan3(x).

Solution

This is a relatively simple integration; the method described below uses a sub-

stitution and the properties sec2 x = 1+tan2 x and tan x dx = − ln | cos x|+c.

tan3 x dx = tan x tan2 x dx (use an identity to reduce degree)

= tan x(sec2 x − 1) dx

= tan x sec2 x dx tan x dx � �� � � �� � −

� u du

= u du − (− ln | cos x| + c)

= 1

2

u2 + ln | cos x| + c

1 2 = 2

tan x + ln | cos x| + c