Q115 / 160 · TS EAMCET 2020 (14 Sep Shift 2) · Mathematics

Assertion $\int_{-a}^a f(x) d x=\int_0^a(f(x)+f(-x)) d x$ Reason (R) $\int_a^b f(x) d x=\int_{g(a)}^{g(b)} f(g(u)) g^{\prime}(u) d u$ The correct option among the following is

  • A
    (A) is true, $(R)$ is true and $(R)$ is the correct explanation for (A)
  • B
    $(A)$ is true, $(R)$ is true but $(R)$ is not the correct explanation for $(A)$
  • Correct answer
    C
    $(A)$ is true but $(R)$ is false
  • D
    (A) is false but (R) is true

Solution

We have, $A=\int_{-a}^a f(x) d x=\int_0^a(f(x)+f(-x)) d x$ A is true, $\quad R=\int_a^b f(x) d x=\int_{g(a)}^{g(b)} f\left(g(u) g^{\prime}(u) d u\right.$ Put, $x=g(u), d x=g(u) d u$ at $x=a=g(u)$ and $x=b \quad u=g^{-1}(b) \Rightarrow u=g^{-1}(a)$ $\therefore \quad \int_a^b f(x) d x=\int_{g^{-1}(a)}^{g^{-1}(b)} f\left(g(x) g^{\prime}(x) d x\right.$ $\therefore \mathrm{R}$ is false.

Hence, the correct answer is option: 'C'.

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