Grade 12Integral Calculus

Integrate the following and answer with full step

Integrate the following and answer with full steps please:-

tan x tan2x tan 3x dx

2 Answers

Arun

Approved Tutor Answer9 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+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

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