Addition, subtraction, multiplication

Calculus Level 2

Evaluate: ∫ x + y x − y t y d t \large \int_{x + y}^{x - y} ty \ dt

where x x and y y are independent of t t .

2 x y 2 2xy^2 − 4 x 2 y -4x^2y 4 x 2 y 4 4x^2y^4 − 2 x y 2 -2xy^2

Ashish Menon by Ashish Menon , 20, India

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2 solutions

Ashish Menon
May 30, 2016

∫ x + y x − y t y d t = y ∫ x + y x − y t d t = y × ( t 2 2 ) ∣ x + y x − y = y × ( ( x − y ) 2 2 − ( x + y ) 2 2 ) = y × ( x 2 + y 2 − 2 x y − x 2 − y 2 − 2 x y 2 ) = y × − 4 x y 2 = − 2 x y 2 \begin{aligned} \LARGE \int_{x + y}^{x - y} ty \ dt & = \LARGE y\int_{x + y}^{x - y} t \ dt\\ \\ & = \LARGE y × \left(\dfrac{t^2}{2}\right){\Huge \vert}_{x + y}^{x - y}\\ \\ & = \LARGE y × \left(\dfrac{{(x - y)}^2}{2} - \dfrac{{(x + y)}^2}{2}\right)\\ \\ & = \LARGE y × \left(\dfrac{x^2 + y^2 - 2xy - x^2 - y^2 - 2xy}{2}\right)\\ \\ & = \LARGE y × \dfrac{-4xy}{2}\\ \\ & = \LARGE \color{#69047E}{\boxed{-2xy^2}} \end{aligned}

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Nice sol.! Did it the same way +1!

Rishabh Tiwari - 5 years ago

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Thanks :) :)

Ashish Menon - 5 years ago

If you don't want any reports , please mention that y y is a constant

Sabhrant Sachan - 5 years ago

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Did it. Thanks :) :)

Ashish Menon - 5 years ago

@Ashish Siva , it shoul be d t dt instead of d x dx .

Chew-Seong Cheong - 4 years, 10 months ago

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Thanks, I corrected the typo.

Ashish Menon - 4 years, 10 months ago

Relevant wiki: Integration Tricks

I = ∫ x + y x − y t y d t Using ∫ a b f ( x ) d x = ∫ a b f ( a + b − x ) d x ⟹ I = 1 2 ∫ x + y x − y ( t y + ( 2 x − t ) y ) d t = ∫ x + y x − y x y d t = x y t ∣ x + y x − y = x y ( x − y − x − y ) = − 2 x y 2 \begin{aligned} I & = \int_{x+y}^{x-y} ty \ dt & \small \color{#3D99F6}{\text{Using }\int^b_a f(x) \ dx = \int^b_a f(a+b - x) \ dx} \\ \implies I & = \frac 12 \int_{x+y}^{x-y} (ty + (2x-t)y) \ dt \\ & = \int _{x+y}^{x-y} xy \ dt \\ & = xyt \ \bigg|_{x+y}^{x-y} \\ & = xy(x-y-x-y) \\ & = \boxed{-2xy^2} \end{aligned}

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