Let's do some calculus! (21)

Calculus Level 3

∫ 0 4 ( x 2 − 4 x + 5 ) sin ⁡ ( x − 2 ) 2 x 2 − 8 x + 11 d x = ? \large \displaystyle \int_{0}^{4} {\dfrac{(x^{2}-4x+5) \sin (x-2)}{2 x^{2} - 8x +11}} \,dx = \, ?

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The answer is 0.

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Tapas Mazumdar by Tapas Mazumdar , 20, India

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2 solutions

Chew-Seong Cheong
Sep 24, 2016

I = ∫ 0 4 ( x 2 − 4 x + 5 ) sin ⁡ ( x − 2 ) 2 x 2 − 8 x + 11 d x By ∫ a b f ( x ) d x = ∫ a b f ( a + b − x ) d x = 1 2 ∫ 0 4 ( ( x 2 − 4 x + 5 ) sin ⁡ ( x − 2 ) 2 x 2 − 8 x + 11 + ( ( 4 − x ) 2 − 4 ( 4 − x ) + 5 ) sin ⁡ ( ( 4 − x ) − 2 ) 2 ( 4 − x ) 2 − 8 ( 4 − x ) + 11 ) d x = 1 2 ∫ 0 4 ( ( x 2 − 4 x + 5 ) sin ⁡ ( x − 2 ) 2 x 2 − 8 x + 11 + ( x 2 − 8 x + 16 − 16 + 4 x + 5 ) sin ⁡ ( 2 − x ) 2 x 2 − 16 x + 32 − 32 + 8 x + 11 ) d x = 1 2 ∫ 0 4 ( ( x 2 − 4 x + 5 ) sin ⁡ ( x − 2 ) 2 x 2 − 8 x + 11 + ( x 2 − 4 x + 5 ) sin ⁡ ( 2 − x ) 2 x 2 − 8 x + 11 ) d x Note that sin ⁡ ( 2 − x ) = − sin ⁡ ( x − 2 ) = 1 2 ∫ 0 4 ( ( x 2 − 4 x + 5 ) sin ⁡ ( x − 2 ) 2 x 2 − 8 x + 11 − ( x 2 − 4 x + 5 ) sin ⁡ ( x − 2 ) 2 x 2 − 8 x + 11 ) d x = 1 2 ∫ 0 4 0 d x = 0 \begin{aligned} I & = \int_0^4 \frac {(x^2-4x+5)\sin(x-2)}{2x^2-8x+11} dx \quad \quad \small \color{#3D99F6}{\text{By }\int_a^b f(x) \ dx = \int_a^b f(a+b-x) \ dx} \\ & = \frac 12 \int_0^4 \left(\frac {(x^2-4x+5)\sin(x-2)}{2x^2-8x+11} + \frac {((4-x)^2-4(4-x)+5)\sin((4-x)-2)}{2(4-x)^2-8(4-x)+11} \right) dx \\ & = \frac 12 \int_0^4 \left(\frac {(x^2-4x+5)\sin(x-2)}{2x^2-8x+11} + \frac {(x^2-8x+16-16+4x+5)\sin(2-x)}{2x^2-16x+32-32+8x+11} \right) dx \\ & = \frac 12 \int_0^4 \left(\frac {(x^2-4x+5)\sin(x-2)}{2x^2-8x+11} + \frac {(x^2-4x+5)\color{#D61F06}{\sin(2-x)}}{2x^2-8x+11} \right) dx \quad \quad \small \color{#D61F06}{\text{Note that }\sin(2-x) = - \sin (x-2)} \\ & = \frac 12 \int_0^4 \left(\frac {(x^2-4x+5)\sin(x-2)}{2x^2-8x+11} \color{#D61F06}{-} \frac {(x^2-4x+5)\color{#D61F06}{\sin(x-2)}}{2x^2-8x+11} \right) dx \\ & = \frac 12 \int_0^4 0 \ dx = \boxed{0} \end{aligned}

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Prince Loomba
Sep 24, 2016

Put x-2=t, change the limits it will become odd function with limits -2,2 or answer is 0

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