The value of $\int_{0}^{2 \pi}[\sin 2 x(1+\cos 3 x)] d x$ where $[t]$ denotes the greatest integer function, is:

JEE Main · 2019 · 10 Apr 2019 · Shift 1 · Definite Integrals

The value of

\int_{0}^{2 π} [sin 2 x (1 + cos 3 x)] d x

where

[t]

denotes the greatest integer function, is:

Answer: (D)

Let

I = \int_{0}^{2 π} ⌊ sin 2 x (1 + cos 3 x)⌋ d x

. Replacing

x

2 π - x

sends

sin 2 x

- sin 2 x

and leaves

cos 3 x

unchanged, so the integrand becomes

⌊ - sin 2 x (1 + cos 3 x)⌋

. Adding the two forms,

2 I = \int_{0}^{2 π} (⌊ t ⌋ + ⌊ - t ⌋) d x

with

t = sin 2 x (1 + cos 3 x)

. Since

t

is non-integer almost everywhere,

⌊ t ⌋ + ⌊ - t ⌋ = - 1

, so

2 I = \int_{0}^{2 π} (- 1) d x = - 2 π

and

I = - π

JEE Main · 2019 · 10 Apr 2019 · Shift 1 · 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 →