The value of the integral $\int_{-1}^{1} \log \left(x+\sqrt{x^{2}+1}\right) d x$ is :

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Definite Integrals

The value of the integral

\int_{- 1}^{1} lo g (x + x^{2} + 1) d x

is :

Answer: (D)

With

f (x) = lo g (x + x^{2} + 1)

f (- x) = lo g (x^{2} + 1 - x) = lo g \frac{1}{x + x ^{2} + 1} = - f (x)

because

(x^{2} + 1 - x) (x^{2} + 1 + x) = 1

. An odd function integrated over the symmetric interval

[- 1, 1]

gives

0

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Single correct option

The value of $\int_{- π /6}^{π /6} (\frac{π + 4 x ^{11}}{1 - s i n ( ∣ x ∣ + π /6 )}) d x$ is equal to
JEE Main 2026View solution →
$6 \int_{0}^{π} ∣ (sin 3 x + sin 2 x + sin x) ∣ d x$ is equal to.......
JEE Main 2026View solution →
Let $[\cdot]$ denote the greatest integer function and $f (x) = lim_{n \to \infty} \frac{1}{n ^{3}} \sum_{k = 1}^{n} [\frac{k ^{2}}{3 ^{x}}]$ . Then $12 \sum_{j = 1}^{\infty} f (j)$ is equal …
JEE Main 2026View solution →
If $\int_{0}^{1} 4 cot^{- 1} (1 - 2 x + 4 x^{2}) dx = a tan^{- 1} (2) - b lo g_{e} (5)$ , where $a, b \in N$ , then $(2 a + b)$ is equal to ____.
JEE Main 2026View solution →
Let $f : [1, \infty) \to R$ be a differentiable function. If $6 \int_{1}^{x} f (t) d t = 3 x f (x) + x^{3} - 4$ for all $x \geq 1$ , then the value of $f (2) - f (3)$ is
JEE Main 2026View solution →
The value of $\int_{- \frac{π}{2}}^{\frac{π}{2}} (\frac{1}{[ x ] + 4}) dx$ , where [•] denotes the greatest integer function, is
JEE Main 2026View solution →