Tricky Calculus Problem

Calculus Level 4

Let f ( x ) = 2 x + sin ( π x 2 ) f(x) = 2x+\sin \left(\dfrac{{\pi}x}{2}\right) and g g be the inverse function of f f .

Compute 3 4 g ( x ) d x \displaystyle \int_3^4 g(x)\,dx .

1.363 1.636 0.682 2.955

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4 solutions

Aaghaz Mahajan
May 24, 2019

Using the beautiful property that

a b f ( x ) d x + f ( a ) f ( b ) f 1 ( x ) d x = b f ( b ) a f ( a ) \displaystyle \int_a^bf\left(x\right)dx+\int_{f\left(a\right)}^{f\left(b\right)}f^{-1}\left(x\right)dx=bf\left(b\right)-af\left(a\right)

We have,

3 4 g ( x ) d x = 5 1 2 ( 2 x + sin ( π x 2 ) ) d x \displaystyle \int_3^4g\left(x\right)dx=5-\int_1^2\left(2x+\sin\left(\frac{\pi x}{2}\right)\right)dx

which simply evaluates to 1.363... 1.363...

Chew-Seong Cheong
May 25, 2019

3 4 g ( x ) d x = g ( 3 ) g ( 4 ) x d f ( x ) Note that f ( x ) = 2 x + sin ( π x 2 ) = 1 2 x ( 2 + π 2 cos ( π x 2 ) ) d x d f ( x ) = ( 2 + π 2 cos ( π x 2 ) ) d x = 1 2 2 x d x + 1 2 π 2 x cos ( π x 2 ) d x By integration by parts = x 2 + x sin ( π x 2 ) sin ( π x 2 ) d x 1 2 = x 2 + x sin ( π x 2 ) + 2 π cos ( π x 2 ) 1 2 = 4 1 + 0 1 2 π 0 = 2 2 π 1.363 \begin{aligned} \int_3^4 g(x) dx & = \int_{g(3)}^{g(4)} x\ df(x) & \small \color{#3D99F6} \text{Note that } f(x) = 2x + \sin \left(\frac {\pi x}2\right) \\ & = \int_1^2 x \left(2 + \frac \pi 2 \cos \left(\frac {\pi x}2\right) \right) dx & \small \color{#3D99F6} \implies df(x) = \left(2 + \frac \pi 2 \cos \left(\frac {\pi x}2\right) \right) dx \\ & = \int_1^2 2x \ dx + \color{#3D99F6} \int_1^2 \frac \pi 2 x \cos \left(\frac {\pi x}2\right) \ dx & \small \color{#3D99F6} \text{By integration by parts} \\ & = x^2 + \color{#3D99F6} x \sin \left(\frac {\pi x}2\right) - \int \sin \left(\frac {\pi x}2\right) dx \ \bigg|_1^2 \\ & = x^2 + x \sin \left(\frac {\pi x}2\right) + \frac 2 \pi \cos \left(\frac {\pi x}2\right) \ \bigg|_1^2 \\ & = 4 - 1 + 0 - 1 - \frac 2 \pi - 0 \\ & = 2 - \frac 2\pi \approx \boxed{1.363} \end{aligned}

Tolga Gürol
May 26, 2019

Doing a numerical integration over a meta-function which computes the inverse of a function numerically gives 1.36338022763213.

Thank you for the identity. See Integral of inverse functions .

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