Integrating Sine Of Cosine

Calculus Level 2

∫ 0 π sin ⁡ ( cos ⁡ x ) d x \large \int_{0}^{\pi} \sin(\cos x) \, dx

Find the value of the closed form of the above integral.

Give your answer to 3 decimal places.


The answer is 0.000.

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Viki Zeta by Viki Zeta , 19, India

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1 solution

Chew-Seong Cheong
Oct 19, 2016

Relevant wiki: Integration Tricks

I = ∫ 0 π sin ⁡ ( cos ⁡ x ) d x Using identity ∫ a b f ( x ) d x = ∫ a b f ( a + b − x ) d x = 1 2 ∫ 0 π sin ⁡ ( cos ⁡ x ) + sin ⁡ ( cos ⁡ ( π − x ) ) d x = 1 2 ∫ 0 π sin ⁡ ( cos ⁡ x ) + sin ⁡ ( − cos ⁡ x ) d x = 1 2 ∫ 0 π sin ⁡ ( cos ⁡ x ) − sin ⁡ ( cos ⁡ x ) d x = 1 2 ∫ 0 π 0 d x = 0 \begin{aligned} I & = \int_0^\pi \sin (\cos x) \ dx & \small \color{#3D99F6}{\text{Using identity }\int_a^b f(x) \ dx = \int_a^b f(a+b-x) \ dx} \\ & = \frac 12 \int_0^\pi \sin (\cos x) + \sin (\cos (\pi-x)) \ dx \\ & = \frac 12 \int_0^\pi \sin (\cos x) + \sin (-\cos x) \ dx \\ & = \frac 12 \int_0^\pi \sin (\cos x) - \sin (\cos x) \ dx \\ & = \frac 12 \int_0^\pi 0 \ dx = \boxed{0} \end{aligned}

Note : The sine function is an odd function, so sin ⁡ ( A ) = − sin ⁡ ( − A ) \sin(A) = -\sin(-A) .

13 Helpful 0 Interesting 0 Brilliant 0 Confused

Just fix that typo

Viki Zeta - 4 years, 8 months ago

Note: We do not have a simple form for the indefinite integral ∫ sin ⁡ ( cos ⁡ x ) \int \sin ( \cos x ) , and it just happens that the parts are symmetric enough to cancel out.

Calvin Lin Staff - 4 years, 7 months ago

You meant sin ⁡ ( − A ) = − s i n A \sin(-A)= - \ sinA

Hana Wehbi - 4 years, 7 months ago

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Isnt' that the same?

There are different ways to write up what it means to be an odd function, but they are equivalent.

Calvin Lin Staff - 4 years, 7 months ago

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Oh yes, you are right. I forgot about that.

Hana Wehbi - 4 years, 7 months ago

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