Integration

Calculus Level 4

Evaluate the following integral ∫ 0 π / 4 e tan ⁡ θ ( sin ⁡ θ − cos ⁡ θ ) 2 d θ . \int_0^{\pi/4} e^{\tan \theta} \left(\sin \theta-\cos \theta\right)^2~\text d \theta. Input your answer up to three decimal places.


The answer is 0.359141.

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Robur 131 by Robur 131 , 34, Bangladesh

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1 solution

Robur 131
Oct 2, 2014

I am writing θ \theta as x x .

e tan ⁡ x ( sin ⁡ x − cos ⁡ x ) 2 = e tan ⁡ x ( 1 − 2 sin ⁡ x cos ⁡ x ) = e tan ⁡ x sec ⁡ 2 x ⋅ cos ⁡ 2 x − 2 sin ⁡ x cos ⁡ x . e^{\tan x} (\sin x -\cos x)^2 = e^{\tan x} (1-2 \sin x \cos x) =e^{\tan x} \sec^2 x \cdot \cos^2 x - 2 \sin x \cos x.

Now if a integral is given in the form e x ( f ( x ) + f ′ ( x ) ) e^x (f(x) + f'(x)) we can write the result as e x f ( x ) + C . e^x f(x) +C.

So ∫ e tan ⁡ x sec ⁡ 2 x ⋅ cos ⁡ 2 x − 2 sin ⁡ x cos ⁡ x = e tan ⁡ x cos ⁡ 2 x + C . \int e^{\tan x} \sec^2 x \cdot \cos^2 x - 2 \sin x \cos x= e^{\tan x} \cos^2 x +C.

Now if we input the values π / 4 \pi/4 and 0 0 we get

e tan ⁡ ( π / 4 ) cos ⁡ 2 ( π / 4 ) − e tan ⁡ 0 cos ⁡ 2 0 = e 1 ⋅ 1 2 − 1 ⋅ 1 = e 2 − 1 = 0.359141 ( Answer ) . e^{\tan(\pi/4)} \cos^2 (\pi/4) - e^{\tan 0} \cos^2 0 = e^1 \cdot \dfrac{1}{2} - 1\cdot 1= \dfrac{e}{2} - 1=0.359141 ~(\text{Answer}).

5 Helpful 0 Interesting 0 Brilliant 0 Confused

you should prove the relation used.

Hasan Kassim - 6 years, 8 months ago

Ya ! actually i used the same concept as you mentioned but i seriously don't understand the way you presented it here.. could you please be a little elaborative .? Please be gentle. - ALIEN

Anurag Pandey - 5 years, 2 months ago

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