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

Sudipan Datta , 8 Years ago
Grade 12
anser 2 Answers
Arun

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+tanB1−tanAtanB

So, we see that:

tan3x=tan(x+2x)=tanx+tan2x1−tanxtan2x

Cross-multiplying:

(1−tanxtan2x)⋅tan3x=tanx+tan2x

Distributing:

tan3x−tanxtan2xtan3x=tanx+tan2x

Solving for tanxtan2xtan3x:

tanxtan2xtan3x=tan3x−tan2x−tanx

Thus:

∫(tanxtan2xtan3x)dx=∫(tan3x−tan2x−tanx)dx

Splitting this apart:

=∫tan3xdx−∫tan2xdx−∫tanxdx

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

Hence a final answer of:

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

ApprovedApproved
Last Activity: 8 Years ago
salman khan
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            
Last Activity: 8 Years ago
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