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Integrate the following and answer with full steps please:- tan x tan2x tan 3x dx

Integrate the following and answer with full steps please:-
tan x tan2x tan 3x dx

Grade:12

2 Answers

Arun
25763 Points
4 years ago

We can rewrite tanxtan2xtan3x in a way that is much easier to integrate. First, find out how else to write tan3x using the tangent angle addition formula.

tan(A+B)=tanA+tanB1tanAtanB

So, we see that:

tan3x=tan(x+2x)=tanx+tan2x1tanxtan2x

Cross-multiplying:

(1tanxtan2x)tan3x=tanx+tan2x

Distributing:

tan3xtanxtan2xtan3x=tanx+tan2x

Solving for tanxtan2xtan3x:

tanxtan2xtan3x=tan3xtan2xtanx

Thus:

(tanxtan2xtan3x)dx=(tan3xtan2xtanx)dx

Splitting this apart:

=tan3xdxtan2xdxtanxdx

Note that tanxdx=ln(|secx|)+C. The first two integrals will require using substitution: let u=3xdu=3dx and v=2xdv=2dx.

Hence a final answer of:

=ln(|sec3x|)3ln(|sec2x|)2ln(|secx|)+C

salman khan
57 Points
3 years ago
we know that  tan3x=tan(2x+x) =(tan2x+tanx)/1-tan2x tanx                                                                                            then cross multiply                    tan3x-tanxtan2xtan3x=tan2x+tanx                                                                                                                  tanxtan2xtan3x=tan3x-tan2x-tanx                                                                                             so      \int tanxtan2xtan3x=\int tan3x-\int tan2x-\int tanx                                                                                                                                       =ln\left | sec3x \right | /3 -ln\left | sec2x \right | /2 -ln\left | secx \right | +c                                            answer.      use this concept of trigonometry to solve other question.                                                                                                                       BEST OF LUCK            

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