Parameterized Integration - Part 2

Calculus Level 3

Given that F ( x ) = ∫ 0 x sin ⁡ 100 ( x − t ) d t F(x) = \displaystyle \int_{0}^{x} \sin^{100}(x-t) \, dt , find d F d x \displaystyle \dfrac{dF}{dx} .

1 1 sin ⁡ 100 x \sin^{100} x 0 0 − sin ⁡ 100 x -\sin^{100} x

Alice Smith by Alice Smith , 20, China

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1 solution

Mark Hennings
Dec 21, 2020

Since F ( x ) = ∫ x 0 sin ⁡ 100 u ( − d u ) = ∫ 0 x sin ⁡ 100 u d u F(x) \; = \; \int_x^0 \sin^{100}u\,(-du) \; =\; \int_0^x \sin^{100}u\,du using the substitution u = x − t u = x -t , we deduce that F ′ ( x ) = sin ⁡ 100 x F'(x) = \boxed{\sin^{100}x} .

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