Trigonometrical Integrals

Calculus Level 3

∫ 0 π 1 1 + tan ⁡ x d x \large \displaystyle \int_{0}^{\pi} \dfrac{1}{1+ \tan x} dx

Evaluate the Cauchy principal value of the above integral. If your answer comes as π A \dfrac{\pi}{A} , then find the value of A A .

Image Credit: Wikimedia ThibautLienart .


The answer is 2.

Anish Harsha by Anish Harsha , 20, India

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1 solution

Pulkit Gupta
Dec 9, 2015

Applying the property f (x) = f ( a + b - x) , we obtain I = {1/ 1 - t a n x \ tan x }

We sum the integrals to obtain 2I = 2 / 1 - t a n x 2 tanx^{2} OR I = 1 - t a n x 2 tanx^{2}

Writing t a n x 2 tanx^{2} as s i n x 2 sinx^{2} / c o s x 2 cosx^{2} ; c o s x 2 cosx^{2} - s i n x 2 sinx^{2} as c o s 2 x \ cos2x ; c o s x 2 cosx^{2} = 1 + cos ⁡ 2 x \cos2x / 2

We simplify the integral to 1 + cos ⁡ 2 x \cos2x / 2 cos ⁡ 2 x \cos2x = sec ⁡ 2 x \sec2x + 1/ 2 ; after which the integral computes to π / 2 \pi/2

4 Helpful 0 Interesting 0 Brilliant 0 Confused

But can you tell me what it has even to do with Cauchy???

Daniel Kazmarek - 8 months ago

Anyways, I learnt about this property at high school. Quite a good move!

Daniel Kazmarek - 8 months ago

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