Aspire Faculty ID #16693 · Topic: CUET 2023 · Just now
CUET 2023

Given below are two statements:

Statement I:
$\displaystyle \int_{-a}^{a} f(x),dx = \int_{0}^{a} [f(x)+f(-x)],dx$

Statement II:
$\displaystyle \int_{0}^{1} \sqrt{(1+x)(1+x^3)},dx \le \dfrac{15}{8}$

In the light of the above statements, choose the most appropriate answer from the options given below:

Solution

Statement I:
This is a standard property of definite integrals.
So Statement I is true.

Statement II:
Using AM ≥ GM:
$(1+x)(1+x^3) \le \left(\dfrac{(1+x)+(1+x^3)}{2}\right)^2$

So,
$\sqrt{(1+x)(1+x^3)} \le \dfrac{2 + x + x^3}{2}$

Integrating from $0$ to $1$:
$\displaystyle \int_0^1 \sqrt{(1+x)(1+x^3)},dx \le \dfrac{15}{8}$

Statement II is true.

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