Girish Mahajan (Editor)

Constant factor rule in integration

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The constant factor rule in integration is a dual of the constant factor rule in differentiation, and is a consequence of the linearity of integration. It states that a constant factor within an integrand can be separated from the integrand and instead multiplied by the integral. For example, where k is a constant:

∫ k d y d x d x = k ∫ d y d x d x .

Proof

Start by noticing that, from the definition of integration as the inverse process of differentiation:

y = ∫ d y d x d x .

Now multiply both sides by a constant k. Since k is a constant it is not dependent on x:

k y = k ∫ d y d x d x . (1)

Take the constant factor rule in differentiation:

d ( k y ) d x = k d y d x .

Integrate with respect to x:

k y = ∫ k d y d x d x . (2)

Now from (1) and (2) we have:

k y = k ∫ d y d x d x k y = ∫ k d y d x d x .

Therefore:

∫ k d y d x d x = k ∫ d y d x d x . (3)

Now make a new differentiable function:

u = d y d x .

Substitute in (3):

∫ k u d x = k ∫ u d x .

Now we can re-substitute y for something different from what it was originally:

y = u .

So:

∫ k y d x = k ∫ y d x .

This is the constant factor rule in integration.

A special case of this, with k=-1, yields:

∫ − y d x = − ∫ y d x .

References

Constant factor rule in integration Wikipedia
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