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Calculus Level 2

∫ sin ⁡ ( x ) d x − ∫ sin ⁡ ( x ) d x = ? \large \int \sin (x) \, dx - \int \sin (x ) \, dx = \ ?

-1 2 An unknown constant 1 0 -2

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Akhil Bansal by Akhil Bansal , 22, India

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7 solutions

∫ sin ⁡ ( x ) d x − ∫ sin ⁡ ( x ) d x = ∫ ( sin ⁡ ( x ) − sin ⁡ ( x ) ) d x = ∫ ( 0 ) d x = C \int\sin(x)\,\mathrm dx-\int\sin(x)\,\mathrm dx\\ =\int\Big(\sin(x)-\sin(x)\Big)\,\mathrm dx\\ =\int\big(0\big)\,\mathrm dx\\ =C\\

63 Helpful 5 Interesting 4 Brilliant 1 Confused
Razik Ridzuan
Aug 5, 2015

Both integration resulting constants C1 and C2. We can't find the exact value of the constant. So, C1 - C2 = C3 in which C3 is a new constant.

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Curtis Clement
Aug 3, 2015

Using the fundamental Theorem of Calculus: d d x ( c o n s t a n t ) = 0 ⟺ ∫ 0 = c o n s t a n t \frac{d}{dx} (constant) = 0 \iff \int 0 = constant

13 Helpful 1 Interesting 0 Brilliant 0 Confused

But if solve it without, you know, combining the 2 given integrals then answer comes out to be 0. How is this possible?

Vandita Soni - 4 years ago

according to basic concept of integration...the integration of any function represents area under the curve of that function...and area cannot be zero...so by convention or taking of derivative of 0 we can say that the best answer is constant

Ammad Ud Din - 5 years, 10 months ago

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Not technically correct because integration refers to the SIGNED area under a curve. Thus, an integral can have a value of 0.

Adam Bachmann - 5 years, 10 months ago
Carlos Arce
Aug 5, 2015

Indefinite integral of sin(x) is -cos(x) + C (constant) So, -cos(x) - (-cos(x)) + C1 + C2 = 0 + C3 = C The C's all add up to another arbitrary constant.

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Lukas Leibfried
Sep 13, 2015

Always remember +c.

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Hadia Qadir
Aug 5, 2015

A constant is the best answer 'cause [d(constant) / d(x)] = 0 hence ∫(sinx - sinx)dx = ∫0 = constant colonthree emoticon

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William G.
Aug 6, 2015

integral of anything is something plus any number therefore any number minus any number is any number. aka constant

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