PYQ Vault
EbooksTutor

Integral of x f(x) using f(x) = f(a + b - x) to relate two integrals

This type of question, asked 3 times: 2 May 2023 S1, 11 May 2024 S1, 11 May 2024 S2. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2 May 2023 S1

    121
    Let ff and gg be continuous functions on [0,a][0,a] such that f(x)=f(a−x)f(x) = f(a-x) and g(x)+g(a−x)=4g(x) + g(a-x) = 4, then ∫0af(x)g(x) dx\displaystyle\int_0^a f(x)g(x)\,dx is equal to

    Tap an option to check your answer.

  2. Question 2

    11 May 2024 S1

    115
    Let f(x)f(x) be positive for all real xx. If I1=∫1−hhxf(x(1−x)) dxI_1 = \int_{1-h}^{h} xf(x(1-x))\,dx and I2=∫1−hhf(x(1−x)) dxI_2 = \int_{1-h}^{h} f(x(1-x))\,dx, where (2h−1)>0(2h-1) > 0, then I1I2\frac{I_1}{I_2} is

    Tap an option to check your answer.

  3. Question 3

    11 May 2024 S2

    136
    Let f:[−1,2]→[0,∞)f:[-1,2]\to[0,\infty) be a continuous function such that f(x)=f(1−x),  ∀x∈[−1,2]f(x)=f(1-x),\;\forall x\in[-1,2]. Let R1=∫−12xf(x) dxR_1=\int_{-1}^{2}xf(x)\,dx and R2R_2 be the area of the region bounded by y=f(x)y=f(x), x=−1x=-1, x=2x=2 and the X-axis, then R2R_2 is

    Tap an option to check your answer.