# Integrals - Class 12 - Mathematics

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## Extra Questions - Integrals | NCERT | Mathematics | Class 12

$\int_{0}^{1} x e^{x^{2}} dx = \lambda \int_{0}^{1} e^{x^{2}} dx$, then

A) $\quad \lambda = 0$

B) $\quad \lambda \in (0,1)$

C) $\quad \lambda \in (-\infty, 0)$

D) $\quad \lambda \in (1,2)$

The correct option is **B**

$$ \lambda \in(0,1) $$

Given the relationships $0 < x < 1$ and $e^{x^2} > 0$, we can deduce:
$$
0 < x e^{x^2} < e^{x^2}
$$
which implies:
$$
\int_0^1 0 , dx < \int_0^1 x e^{x^2} , dx < \int_0^1 e^{x^2} , dx
$$
Given the equation:
$$
\int_{0}^{1} x e^{x^{2}} dx = \lambda \int_{0}^{1} e^{x^{2}} dx
$$
The inequality transforms to:
$$
0 < \lambda \int_{0}^{1} e^{x^{2}} dx < \int_{0}^{1} e^{x^{2}} dx
$$
After dividing through by $\int_{0}^{1} e^{x^{2}} dx$ (which is positive), we get:
$$
0 < \lambda < 1
$$
Hence, **$\lambda$ belongs to the interval $(0,1)$**.

Find the period of a real-valued function satisfying $$ f(x) + f(x + 4) = f(x + 2) + f(x + 6) $$

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